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Q. Let $\mathrm{f}(\mathrm{x})=\mathrm{x}|\mathrm{x}|$ and $\mathrm{g}(\mathrm{x})=\sin \mathrm{x}$
Statement–1 : gof is differentiable at x = 0 and its derivative is continuous at that point.
Statement–2 : gof is twice differentiable at x = 0.
(1) Statement–1 is true, Statement–2 is false.
(2) Statement–1 is false, Statement–2 is true.
(3) Statement–1 is true, Statement–2 is true; Statement–2 is a correct explanation for Statement– 1.
(4) Statement–1 is true, Statement–2 is true ; Statement–2 is not a correct explanation for statement– 1.
[AIEEE-2009]
Ans. (1)
$F(x)=x|x| \quad g(x)=\sin x$
$\operatorname{gof}=\sin x|x|=\left[\begin{array}{cc}{2 x \cos x^{2}} & {x>0} \\ {-2 x \cos x^{2}} & {x<0}\end{array}\right.$
$(\mathrm{gof})^{\prime \prime}=\left[\begin{array}{cc}{2 \cos \mathrm{x}^{2}-4 \mathrm{x}^{2} \sin \mathrm{x}^{2}} & {\mathrm{x}>0} \\ {-2 \cos \mathrm{x}^{2}+4 \mathrm{x}^{2} \sin \mathrm{x}^{2}} & {\mathrm{x}<0}\end{array}\right.$
$\Rightarrow$ First derivative exist but second derivative doesn't exist.
Q. If function $\mathrm{f}(\mathrm{x})$ is differentiable at $\mathrm{x}=\mathrm{a}$ then $\lim _{x \rightarrow a} \frac{\mathrm{x}^{2} \mathrm{f}(\mathrm{a})-\mathrm{a}^{2} \mathrm{f}(\mathrm{x})}{\mathrm{x}-\mathrm{a}}$
(1) $2 \mathrm{a} \mathrm{f}(\mathrm{a})+\mathrm{a}^{2} \mathrm{f}^{\prime}(\mathrm{a})$
(2) $-a^{2} f^{\prime}(a)$
(3) af $(a)-a^{2} f^{\prime}(a)$
(4) $2 \mathrm{af}(\mathrm{a})-\mathrm{a}^{2} \mathrm{f}^{\prime}(\mathrm{a})$
[AIEEE-2011]
Ans. (4)
Q. Consider the function, $f(x)=|x-2|+|x-5|, x \in R .$
Statement–1 : f'(4) = 0.
Statement–2 : f is continuous in [2, 5], differentiable in (2, 5) and f(2) = f(5).
[AIEEE 2012]
(1) Statement–1 is true, Statement–2 is false.
(2) Statement–1 is false, Statement–2 is true.
(3) Statement–1 is true, Statement–2 is true ; Statement–2 is a correct explanation for Statement1.
(4) Statement–1 is true, Statement–2 is true ; Statement–2 is not a correct explanation for Statement1.
[AIEEE 2012]
Ans. (4)
Q. Let $\mathrm{f}(\mathrm{x})=\mathrm{x}|\mathrm{x}|, \mathrm{g}(\mathrm{x})=\sin \mathrm{x}$ and $\mathrm{h}(\mathrm{x})=(\mathrm{gof})(\mathrm{x}) .$ Then
(1) h(x) is differentiable at x = 0
(2) h(x) is continuous at x = 0 but is not differentiable at x = 0
(3) h(x) is differentiable at x = 0 but h(x) is not continuous at x = 0
(4) h(x) is not differentiable at x = 0
[On-line 2014]
Ans. (2)
Q. Let $\mathrm{f}, \mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}$ be two functions defined byf(x) $=\left\{\begin{array}{ll}{\mathrm{x} \sin \left(\frac{1}{\mathrm{x}}\right), \mathrm{x} \neq 0} & {, \text { and } \mathrm{g}(\mathrm{x})=\mathrm{xf}(\mathrm{x}):-} \\ {0} & {, \mathrm{x}=0}\end{array}\right.$
Statement I : f is a continuous function at x = 0.
Statement II : g is a differentiable function at x = 0.
(1) Statement I is false and statement II is true
(2) Statement I is true and statement II is false
(3) Both statement I and II are true
(4) Both statements I and II are false
[on-line 2014]
Ans. (3)
Q. Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be a function such that $|\mathrm{f}(\mathrm{x})| \leq \mathrm{x}^{2},$ for all $\mathrm{x} \in \mathrm{R} .$ Then, at $\mathrm{x}=0, \mathrm{f}$ is:
(1) Neither continuous nor differentiable
(2) differentiable but not continuous
(3) continuous as well as differentiable
(4) continuous but not differentiable
[on-line 2014]
Ans. (3)
$|\mathrm{f}(\mathrm{x})| \leq \mathrm{x}^{2}$
$-\mathrm{x}^{2} \leq \mathrm{f}(\mathrm{x}) \leq \mathrm{x}^{2} \therefore$ at $\mathrm{x}=0 \mathrm{f}(\mathrm{x})$ is continuons as well as differentiable
Q. For $\mathrm{x} \in \mathrm{R}, \mathrm{f}(\mathrm{x})=|\log 2-\sin \mathrm{x}|$ and $\mathrm{g}(\mathrm{x})=\mathrm{f}(\mathrm{f}(\mathrm{x})),$ then :
(1) $\mathrm{g}$ is differentiable at $\mathrm{x}=0$ and $\mathrm{g}^{\prime}(0)=-\sin (\log 2)$
(2) $\mathrm{g}$ is not differentiable at $\mathrm{x}=0$
(3) $\mathrm{g}^{\prime}(0)=\cos (\log 2)$
(4) $\mathrm{g}^{\prime}(0)=-\cos (\log 2)$
[JEE Mains 2016]
Ans. (3)
Q. Let $\mathrm{S}=\left\{\mathrm{t} \in \mathrm{R}: \mathrm{f}(\mathrm{x})=|\mathrm{x}-\pi| \cdot\left(\mathrm{e}^{|\mathrm{x}|}-1\right)\right.$ $\sin |x|$ is not differentiable at t}. Then the set S is equal to:
(1) {0} (2) $\{\pi\}$ (3) $\{0, \pi\}$ (4) $\phi$ (an empty set)
[JEE Mains 2018]
Ans. (4)
Q. Let $\mathrm{S}=\{\lambda, \mu)$ $\varepsilon \operatorname{RxR}: f(t)$ $\left.=(|\lambda|) \mathrm{e}^{|t|}-\mu\right)$. $\sin (2|\mathrm{t}|)$, $\mathrm{t} \varepsilon \mathrm{R},$ is a differentiable function}.Then S is a subset of :
(1) $[0, \infty) \times \mathrm{R}$
(2) $\mathrm{R} \times(-\infty, 0)$
(3) $\mathrm{R} \times[0, \infty)$
(4) $(-\infty, 0) \times \mathrm{R}$
[JEE Mains 2018]
Ans. (3)