JEE Main & Advanced Mathematics Functions Question Bank

done Differentiability Total Questions - 48

• question_answer1)
  Which of the following statements is true
  A)
             A continuous function is an increasing function done clear
  B)
             An increasing function is continuous done clear
  C)
             A continuous function is differentiable done clear
  D)
             A differentiable function is continuous done clear
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• question_answer2)
  If \[f(x)=\left\{ \begin{align}   & \ x+1,\ \text{when}\,x<2 \\  & 2x-1,\text{when }x\ge \text{2} \\ \end{align} \right.\], then \[f'(2)\] equals [MP PET 1997]
  A)
             0 done clear
  B)
             1 done clear
  C)
             2 done clear
  D)
             Does not exist done clear
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• question_answer3)
  If \[f(x)=\left\{ \begin{align}   & x\frac{{{e}^{(1/x)}}-{{e}^{(-1/x)}}}{{{e}^{(1/x)}}+{{e}^{(-1/x)}}},\,x\ne 0 \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0,\,x=0 \\ \end{align} \right.\] then which of the following is true                        [Kurukshetra CEE 1998]
  A)
             f is continuous and differentiable at every point done clear
  B)
             f is continuous at every point but is not differentiable done clear
  C)
             f is differentiable at every point done clear
  D)
             f is differentiable only at the origin done clear
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• question_answer4)
  If \[f(x)=|x-3|,\]then f is [SCRA 1996; RPET 1997]
  A)
             Discontinuous at \[x=2\]           done clear
  B)
             Not differentiable \[x=2\] done clear
  C)
             Differentiable at \[x=3\]           done clear
  D)
             Continuous but not differentiable at \[x=3\] done clear
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• question_answer5)
  Let \[h(x)=\min \{x,\,{{x}^{2}}\},\]for every real number of x. Then [IIT 1998]
  A)
             h is continuous for all x done clear
  B)
             h is differentiable for all x         done clear
  C)
             \[h'(x)=1\], for all \[b=1\] done clear
  D)
             h is not differentiable at two values of x done clear
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• question_answer6)
  There exists a function \[f(x)\]satisfying \[f(0)=1\], \[f'(0)=-1,\ f(x)>0\]for all x and                    [Kurukshetra CEE 1998]
  A)
             \[f(x)<0\],\[\forall x\] done clear
  B)
             \[-1<f''(x)<0,\,\forall x\] done clear
  C)
             \[-2<f''(x)\le -1,\,\forall x\] done clear
  D)
             \[f''(x)<-2,\,\forall x\] done clear
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• question_answer7)
  The function \[f(x)=\left\{ \begin{align}   & x,\,\,\text{if 0}\le x\le \text{1} \\  & \text{1,}\,\text{ if}\,1<x\le 2 \\ \end{align} \right.\] is                                                                              [SCRA 1996]
  A)
             Continuous at all x, \[0\le x\le 2\]and differentiable at all x, except \[2/3\]in the interval [0,2] done clear
  B)
             Continuous and differentiable at all x in [0,2] done clear
  C)
             Not continuous at any point in [0,2] done clear
  D)
             Not differentiable at any point [0,2] done clear
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• question_answer8)
  The function \[f(x)=|x|\] at \[x=0\] is [MP PET 1993]
  A)
             Continuous but non-differentiable done clear
  B)
             Discontinuous and differentiable done clear
  C)
             Discontinuous and non-differentiable done clear
  D)
             Continuous and differentiable done clear
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• question_answer9)
  Consider \[f(x)=\left\{ \begin{align}   & \frac{{{x}^{2}}}{|x|},\,x\ne 0 \\  & \,\,\,\,\,\,\,0,\,x=0 \\ \end{align} \right.\] [EAMCET 1994]
  A)
             \[f(x)\]is discontinuous everywhere done clear
  B)
             \[f(x)\]is continuous everywhere done clear
  C)
             \[f'(x)\]exists in \[(-1,1)\] done clear
  D)
             \[f'(x)\]exists in \[(-2,2)\] done clear
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• question_answer10)
  At the point \[x=1\], the given function \[f(x)=\left\{ \begin{align}   & {{x}^{3}}-1;\,\,1<x<\infty  \\  & x-1;\,\,-\infty <x\le 1 \\ \end{align} \right.\] is [Roorkee 1993]
  A)
             Continuous and differentiable done clear
  B)
             Continuous and not differentiable done clear
  C)
             Discontinuous and differentiable done clear
  D)
             Discontinuous and not differentiable done clear
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• question_answer11)
  Let \[[x]\]denotes the greatest integer less than or equal to x. If \[f(x)=[x\sin \pi x]\], then \[f(x)\]is [IIT 1986]
  A)
             Continuous at \[x=0\] done clear
  B)
             Continuous in \[(-1,0)\] done clear
  C)
             Differentiable in (?1,1) done clear
  D)
             All the above done clear
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• question_answer12)
  The function defined by \[f(x)=\left\{ \begin{align}   & |x-3|\,;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\ge 1 \\  & \frac{1}{4}{{x}^{2}}-\frac{3}{2}x+\frac{13}{4};\,x<1 \\ \end{align} \right.\] is [IIT 1988]
  A)
                     Continuous at \[x=1\] done clear
  B)
             Continuous at \[x=3\] done clear
  C)
             Differentiable at \[x=1\] done clear
  D)
             All the above done clear
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• question_answer13)
  If \[f(x)=\left\{ \begin{matrix}    {{e}^{x}}+ax, & x<0  \\    b{{(x-1)}^{2}}, & x\ge 0  \\ \end{matrix} \right.\] is differentiable at \[x=0,\] then \[(a,\,b)\] is                  [MP PET 2000]
  A)
             \[(-3,\,-1)\] done clear
  B)
             \[(-3,\,\,1)\] done clear
  C)
             \[(3,\,\,1)\] done clear
  D)
             \[(3,\,-1)\] done clear
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• question_answer14)
  The function \[y\,=\,|\sin x|\] is continuous for any x but it is not differentiable at [AMU 2000]
  A)
             \[x=0\] only                                done clear
  B)
             \[x=\pi \] only done clear
  C)
             \[x=k\,\pi \,(k\] is an integer) only done clear
  D)
             \[x=0\] and \[x=k\,\pi \,(k\] is an integer) done clear
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• question_answer15)
  The function \[y={{e}^{-|x|}}\] is [AMU 2000]
  A)
             Continuous and differentiable at \[x=0\] done clear
  B)
             Neither continuous nor differentiable at \[x=0\] done clear
  C)
             Continuous but not differentiable at \[x=0\] done clear
  D)
             Not continuous but differentiable at \[x=0\] done clear
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• question_answer16)
  A function \[f(x)\,=\left\{ \begin{matrix}    1+x, & x\le 2  \\    5-x, & x>2  \\ \end{matrix} \right.\,\] is [AMU 2001]
  A)
             Not continuous at \[x=2\] done clear
  B)
             Differentiable at \[x=2\] done clear
  C)
             Continuous but not differentiable at \[x=2\] done clear
  D)
             None of these done clear
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• question_answer17)
  The left-hand derivative of \[f(x)=[x]\sin (\pi x)\]  at \[x=k,\,\,k\]is an integer and \[[x]\]= greatest integer \[\le x,\,\] is [IIT Screening 2001]
  A)
             \[{{(-1)}^{k}}\,\,(k-1)\,\pi \] done clear
  B)
             \[{{(-1)}^{k-1}}(k-1)\,\pi \] done clear
  C)
             \[{{(-1)}^{k}}k\pi \] done clear
  D)
             \[{{(-1)}^{k-1}}k\,\pi \] done clear
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• question_answer18)
  Let \[f(x)\,=\left\{ \begin{matrix}    x+1, & \text{when} & x<2  \\    2x-1, & \text{when} & x\ge 2  \\ \end{matrix} \right.\,,\,\] then \[{f}'(2)=\] [Karnataka CET 2002]
  A)
             0 done clear
  B)
             1 done clear
  C)
             2 done clear
  D)
             Does not exist done clear
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• question_answer19)
  Let \[f(x)=\left\{ \begin{matrix}    0, & x<0  \\    {{x}^{2}}, & x\ge 0  \\ \end{matrix} \right.\] , then for all values of \[x\] [IIT 1984; MP PET 2002]
  A)
              f is continuous but not differentiable done clear
  B)
              f  is differentiable but not continuous done clear
  C)
             \[{f}'\] is continuous but not differentiable done clear
  D)
             \[{f}'\] is continuous and differentiable done clear
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• question_answer20)
  sThe function \[f(x)=\left\{ \begin{matrix}    {{e}^{2x}}-1 & , & x\le 0  \\    ax+\frac{b{{x}^{2}}}{2}-1 & , & x>0  \\ \end{matrix} \right.\] is continuous and differentiable for [AMU 2002]
  A)
             \[[.]\] done clear
  B)
             \[a=2,\,b=4\] done clear
  C)
             \[a=2,\,\]any \[b\] done clear
  D)
             Any \[a,\,\,\,b=4\] done clear
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• question_answer21)
  Which of the following is not true       [Kerala (Engg.) 2002]
  A)
             A polynomial function is always continuous done clear
  B)
             A continuous function is always differentiable done clear
  C)
             A differentiable function is always continuous done clear
  D)
                                                                                                                   is continuous for all x  done clear
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• question_answer22)
  The function                                                           \[f(x)={{x}^{2}}\,\,\sin \frac{1}{x},\,x\ne \,0,\,\,f(0)\,=0\]  at                                                                                                   \[x=0\] [MP PET 2003]
  A)
             Is continuous but not differentiable done clear
  B)
             Is discontinuous done clear
  C)
             Is having continuous derivative done clear
  D)
             Is continuous and differentiable done clear
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• question_answer23)
  If                                                                              \[f(x)\,=\,\,\left\{ \begin{matrix}    \frac{x-1}{2{{x}^{2}}-7x+5} & \text{for }x\ne 1  \\    -\frac{1}{3} & \text{for }x=1  \\ \end{matrix}\,\,, \right.\]  then                                                                                                  \[f'(1)=\] [EAMCET 2003]
  A)
             ?1/9 done clear
  B)
             ?2/9 done clear
  C)
             ?1/3 done clear
  D)
             1/3 done clear
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• question_answer24)
  If                                                                                     \[f(x)\,=\frac{x}{1+|x|}\]  for                                                                                                 \[x\in R,\]  then                                                                                                  \[f'(0)=\]                    [EAMCET 2003]
  A)
             0 done clear
  B)
             1 done clear
  C)
             2 done clear
  D)
             3 done clear
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• question_answer25)
  The value of m for which the function                                                                                   \[f(x)=\left\{ \begin{align}   & m{{x}^{2}},\,x\le 1 \\  & \,\,\,\,2x,\,x>1 \\ \end{align} \right.\]  is differentiable at                                                                                                   \[x=1\] ,is [MP PET 1998]
  A)
             0 done clear
  B)
             1 done clear
  C)
             2 done clear
  D)
             Does not exist done clear
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• question_answer26)
  Let                                                                                  \[f(x)=\left\{ \begin{matrix}    \,\,\,\,\,\,\,\,\sin x, & \text{for }x\ge 0  \\    1-\cos x, & \text{for }x\le 0  \\ \end{matrix} \right.\]  and                                                                                           \[g(x)={{e}^{x}}\] . Then                                                                                               \[(gof)'(0)\]  is [UPSEAT 2004]
  A)
             1 done clear
  B)
             ?1 done clear
  C)
             0 done clear
  D)
             None of these done clear
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• question_answer27)
  Suppose                                                                                                   \[f(x)\]  is differentiable at                                                                                                   \[x=1\]  and                                                         \[\underset{h\to 0}{\mathop{\lim }}\,\frac{1}{h}f(1+h)=5\] , then                                                                                                   \[f'(1)\]  equals [AIEEE 2005]
  A)
             5 done clear
  B)
             6 done clear
  C)
             3 done clear
  D)
             4 done clear
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• question_answer28)
  If f is a real- valued differentiable function satisfying                                                                           \[|f(x)-f(y)|\le {{(x-y)}^{2}},x,y\in R\]  and                                                                                                 \[f(0)=0\] , then                                                                                                   \[f(1)\]  equal          [AIEEE 2005]
  A)
             2 done clear
  B)
             1 done clear
  C)
             ?1 done clear
  D)
             0 done clear
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• question_answer29)
  Let                                                                                                     \[f\]  be differentiable for all                                                                                                     \[x\] . If                                                                                                \[f(1)=-2\]  and                                                                                              \[f'(x)\ge 2\]  for                                                                                               \[x\in [1,6]\] , then [AIEEE 2005]
  A)
                                                                                                  \[f(6)<5\] done clear
  B)
                                                                                                  \[f(6)=5\] done clear
  C)
                                                                                                \[f(6)\ge 8\] done clear
  D)
                                                                                                  \[f(6)<8\] done clear
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• question_answer30)
                                                                          \[f(x)=\left| \left| x \right|-1 \right|\] is not differentiable at [IIT Screening 2005]
  A)
             0 done clear
  B)
                                                                                               \[\pm 1,\,0\] done clear
  C)
             1 done clear
  D)
                                                                                                 \[\pm \,1\] done clear
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• question_answer31)
  If \[f(x)\] is twice differentiable polynomial function such that \[f(1)=1,f(2)=-4,f(3)=9\], then [IIT Screening 2005]
  A)
             \[f''(x)=2,\forall x\in R\]                   done clear
  B)
             There exist at least one \[x\in (1,\,3)\] such that \[f''(x)=2\] done clear
  C)
                     There exist at least one \[x\in (2,\,3)\] such that  \[f'(x)=5=f''(x)\]  done clear
  D)
             There exist at least one \[x\in (1,\,2)\] such that \[f(x)=3\] done clear
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• question_answer32)
  If \[f(x)\] is a differentiable function such that \[f:R\to r\] and \[f\left( \frac{1}{n} \right)=0\ \forall \ n\ge 1,n\in I\] then [IIT Screening 2005]
  A)
             \[f(x)=0\ \forall \ x\in (0,\,1)\] done clear
  B)
             \[f(x)=0\forall x\in (0,\,1)\] done clear
  C)
             \[f(0)=0\] but \[f'(0)\] may or may not be 0 done clear
  D)
             \[|f(x)|\,\le 1\ \forall \ x\in (0,\,1)\] done clear
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• question_answer33)
  Let f be continuous on [1, 5] and differentiable in (1, 5). If \[f(1)\]=?3 and \[f'(x)\ge 9\] for all \[x\in (1,\ 5)\], then [Kerala (Engg.) 2005]
  A)
             \[f(5)\ge 33\] done clear
  B)
             \[f(5)\ge 36\] done clear
  C)
             \[f(5)\le 36\] done clear
  D)
             \[f(5)\ge 9\] done clear
  E)
             \[f(5)\le 9\] done clear
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• question_answer34)
  Let \[f(x+y)=f(x)f(y)\] and \[f(x)=1+\sin (3x)g(x)\] where \[g(x)\] is continuous then \[f'(x)\] is [Kerala (Engg.) 2005]
  A)
             \[f(x)g(0)\] done clear
  B)
             \[3g(0)\] done clear
  C)
             \[f(x)\cos 3x\] done clear
  D)
             \[3f(x)g(0)\] done clear
  E)
             \[3f(x)g(x)\] done clear
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• question_answer35)
  Let \[f(x)=\left\{ \begin{align}   & 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall x<0 \\  & 1+\sin x\,\,\,\forall 0\le x\le \pi /2 \\ \end{align} \right.\], then what is the value of \[f'(x)\] at \[x=0\] [Orissa JEE 2005]
  A)
             1 done clear
  B)
             ?1 done clear
  C)
             \[\infty \] done clear
  D)
             does not exist done clear
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• question_answer36)
  If  \[f(x)={{x}^{2}}-2x+4\] and \[\frac{f(5)-f(1)}{5-1}=f'(c)\] then value of  c will be [AMU 2005]
  A)
             0 done clear
  B)
             1 done clear
  C)
             2 done clear
  D)
             3 done clear
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• question_answer37)
  Let \[f(x+y)=f(x)+f(y)\]and \[f(x)={{x}^{2}}g(x)\] for all \[x,y\in R\], where \[g(x)\] is continuous function. Then \[f'(x)\] is equal to
  A)
             \[g'(x)\] done clear
  B)
             \[g(0)\] done clear
  C)
             \[g(0)+g'(x)\] done clear
  D)
             0 done clear
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• question_answer38)
  The function \[f(x)=({{x}^{2}}-1)|{{x}^{2}}-3x+2|+\cos (|x|)\] is not differentiable at [IIT 1999]
  A)
             ?1 done clear
  B)
             0 done clear
  C)
             1 done clear
  D)
             2 done clear
  View Solution play_arrow

• question_answer39)
  The function which is continuous for all real values of x and differentiable at \[x=0\]is [MP PET 1996]
  A)
             \[|x|\] done clear
  B)
             \[\log x\] done clear
  C)
             sin x done clear
  D)
             \[{{x}^{\frac{1}{2}}}\] done clear
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• question_answer40)
  Which of the following is not true [Kurukshetra CEE 1996]
  A)
             Every differentiable function is continuous done clear
  B)
             If derivative of a function is zero at all points, then the function is constant done clear
  C)
             If a function has maximum or minima at a point, then the function is differentiable at that point and its derivative is zero done clear
  D)
             If a function is constant, then its derivative is zero at all points done clear
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• question_answer41)
  If \[f(x)=\left\{ \begin{align}   & x+2\,,-1<x<3 \\  & 5\,\,\,\,\,\,\,\,,\,\,\,\,x=3 \\  & 8-x\,,\,\,\,\,x>3 \\ \end{align} \right.\], then at \[x=3\], \[f'(x)=\] [MP PET 2001]           
  A)
             1 done clear
  B)
             ? 1 done clear
  C)
             0 done clear
  D)
             Does not exist done clear
  View Solution play_arrow

• question_answer42)
  If \[f(x)=\left\{ \begin{align}   & x,\,\,\,\,\,\,\,\,\,\,\,0\le x\le 1 \\  & 2x-1,\,\,\,1<x \\ \end{align} \right.\], then [Orissa JEE 2002]
  A)
             f is discontinuous at\[x=1\] done clear
  B)
             f is differentiable at \[x=1\] done clear
  C)
             f is continuous but not differentiable at \[x=1\]                          done clear
  D)
             None of these done clear
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• question_answer43)
  If \[f(x)=\left\{ \begin{align}   & \,\,\,\,\,\,\,\,\,\,\,\,1,\,\,x<0 \\  & 1+\sin x,\,\,0\le x<\frac{\pi }{2} \\ \end{align} \right.\]then \[f'(0)=\] [MP PET 1994]
  A)
             1 done clear
  B)
             0 done clear
  C)
             \[\infty \] done clear
  D)
             Does not exist done clear
  View Solution play_arrow

• question_answer44)
  If \[f(x)=\left\{ \begin{align}   & a{{x}^{2}}+b;\,\,x\le 0 \\  & \,\,\,\,\,\,\,\,\,{{x}^{2}};x>0\, \\ \end{align} \right.\] possesses derivative at \[x=0\], then
  A)
             \[a=0,b=0\]                                done clear
  B)
             \[a>0,=0\] done clear
  C)
             \[a\in R,=0\] done clear
  D)
             None of these done clear
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• question_answer45)
  The set of all those points, where the function \[f(x)=\frac{x}{1+|x|}\]is differentiable, is
  A)
             \[(-\infty ,\infty )\] done clear
  B)
             \[[0,\infty ]\] done clear
  C)
             \[(-\infty ,\,0)\cup (0,\infty )\] done clear
  D)
             \[(0,\infty )\] done clear
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• question_answer46)
  Function \[y={{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right)\] is not differentiable for [IIT Screening]           
  A)
             \[|x|\,<1\] done clear
  B)
             \[x=1,-1\] done clear
  C)
             \[|x|\,>1\] done clear
  D)
             None of these done clear
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• question_answer47)
  If \[f(x)=x(\sqrt{x}-\sqrt{x+1}),\] then  [IIT 1985]
  A)
             \[f(x)\] is continuous but non- differentiable at \[x=0\] done clear
  B)
             \[f(x)\] is differentiable at \[x=0\] done clear
  C)
             \[f(x)\] is not differentiable at \[x=0\]                                                   done clear
  D)
             None of these done clear
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• question_answer48)
  The number of points at which the function \[f(x)=|x-0.5|+|x-1|+\tan x\] does not have a derivative in the interval (0, 2), is [MNR 1995]
  A)
             1 done clear
  B)
             2 done clear
  C)
                  3 done clear
  D)
                  4 done clear
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