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question_answer1) If \[f\left( x \right)\]be a continuous function defined for \[1\le \,\,x\le \,\,3,\,f\left( x \right)\in \,Q\,\,\forall \,\,x\in \left[ 1,3 \right],\,\,\,f\left( 2 \right)=10,\]then find value of \[f\left( 1.8 \right)\](where Q is a set of all rational numbers).
question_answer2) If the function \[f(x)=\left\{ \begin{matrix} \frac{\sin \sqrt[3]{x}\,\log \,(1+3x)}{{{({{\tan }^{-1}}\sqrt{x})}^{2}}({{e}^{5\sqrt[3]{x}}}-1)}, & x\ne 0 \\ a, & x=0 \\ \end{matrix}, \right.\] is continuous at \[x=0\] then find a.
question_answer3) Let \[f\left( x= \right)\left\{ \begin{matrix} \frac{\begin{matrix} \left( 3/{{x}^{2}} \right)\sin \,2{{x}^{2}} \\ {{x}^{2}}+2x+c \\ \end{matrix}}{1-3{{x}^{2}}} \\ 0 \\ \end{matrix} \right.\begin{matrix} x<0 \\ x\,\,\underline{>}\,\,0,\,x\ne \frac{1}{\sqrt{3}}. \\ x=\frac{1}{\sqrt{3}} \\ \end{matrix}\] If f be continuous at\[x=0\], then find value of c.
question_answer4) Find the value of p, for which \[f\left( x \right)=\left\{ \begin{matrix} \frac{{{\left( {{4}^{x}}-1 \right)}^{3}}}{\sin \left( \frac{x}{p} \right){{\log }_{e}}\left\{ 1+\left( \frac{{{x}^{2}}}{3} \right) \right\}}, & x\ne 0 \\ 12{{\left( {{\log }_{e}}4 \right)}^{3}}, & x=0 \\ \end{matrix} \right.\] is continuous at \[x=0.\]
question_answer5) If \[\left\{ \begin{matrix} \frac{1-\sqrt{2}\,\sin \,x}{\pi -4x}; & x\ne \frac{\pi }{4} \\ a; & x=\frac{\pi }{4} \\ \end{matrix} \right.\] is continuous at \[x=\frac{\pi }{4}\] then find value of \[a\].
question_answer6) If \[f\left( x \right)=\left\{ \begin{matrix} \frac{{{2}^{x}}-1}{\sqrt{1+x}-1}, & -1\le \,x<\infty ,x\ne 0\text{ }\!\!~\!\!\text{ } \\ \text{ }k, & x=0\text{ }\!\!~\!\!\text{ } \\ \end{matrix} \right.\] is continuous everywhere, then k is equal to \[{{\log }_{e}}\lambda \] then find \[\lambda \].
question_answer7) Find the value of\[f\left( 0 \right)\], so that the function \[f\left( x \right)=\frac{2x-{{\sin }^{-1}}x}{2x+{{\tan }^{-1}}x}\]is continuous at each point in its domain.
question_answer8) Let \[f\left( x \right)={{\left( \sin x \right)}^{\frac{1}{\pi -2x}}},x\ne \frac{\pi }{2}\]If \[f\left( x \right)\]is continuous at \[x=\frac{\pi }{2}\]then find the value of\[f\left( \frac{\pi }{2} \right)\].
question_answer9) Find the number of point at which function \[f\left( x \right)=\left| x-0.5 \right|+\left| x-1 \right|+\tan \,\,x,\]does not have a derivation in the interval \[\left( 0,2 \right)\]
question_answer10) Find the value of b such that the function \[f\left( x \right)=\left\{ \begin{matrix} ax+3, & x\ge 1\text{ }\!\!~\!\!\text{ } \\ {{x}^{2}}+b, & x<1\text{ }\!\!~\!\!\text{ } \\ \end{matrix} \right.\] is continuous and differentiable at\[x=1\].
question_answer11) If is differentiable for \[x\in R\] then find value of \[\left( a-\frac{b}{e} \right).\]
question_answer12) Find number of non-differentiable point for \[f\left( x \right)=\min \left( \sin x,\,\,\cos x \right)\]is\[\left( if\,\,x\in \left( 0,\,4\pi \right) \right)\].
question_answer13) Find the number of point where \[f\left( x \right)=\left| \,\,\,\left| x \right|-1 \right|\]is not differentiable.
question_answer14) If the function \[g\left( x \right)=\left\{ \begin{matrix} k\sqrt{x+1}, & 0\le x\le 3 \\ mx+2, & 3<x\le 5 \\ \end{matrix} \right.\]is differentiable, then find the value of\[k+m\].
question_answer15) Suppose \[f\left( x \right)\] is differentiable at \[x=1\] and \[\underset{h\to 0}{\mathop{\lim }}\,\frac{1}{h}f\left( 1+h \right)=5,\] then find \[f'\left( 1 \right)\].
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is differentiable for \[x\in R\] then find value of \[\left( a-\frac{b}{e} \right).\] 