-
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)
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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
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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
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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
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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
View Solution play_arrow