MCQ
If $f(x) = \,|x|,$ then $f'(0) = $
- A$0$
- B$1$
- C$x$
- ✓None of these
$f(0 + 0) = \mathop {\lim }\limits_{h \to 0} |0 + h| = 0$ and $f(0 - 0) = \mathop {\lim }\limits_{h \to 0} |0 - h| = 0$
$Rf'(0) = \mathop {\lim }\limits_{h \to 0} \frac{{f(0 + h) - f(0)}}{h} = \mathop {\lim }\limits_{h \to 0} \frac{{|h| - 0}}{h}$
$ = \mathop {\lim }\limits_{h \to 0} \frac{h}{h}\,\,(h$ being positive $)=1$
$Lf'(0) = \mathop {\lim }\limits_{h \to 0} \frac{{f(0 - h) - f(0)}}{{ - h}} = \mathop {\lim }\limits_{h \to 0} \frac{{|h| - 0}}{{ - h}}$
$ = \mathop {\lim }\limits_{h \to 0} \frac{h}{{ - h}} \,\, (h$ being positive $) = -1.$
$\therefore Rf'(0) \ne Lf'(0)$.
The function $f$ is not differentiable.
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$I$. $A(t) < 0$ for all $t$.
$II$. $A(t)$ has infinitely many critical points.
$III.$ $A(t)=0$ for infinitely many $t$.
$IV$. $A^{\prime}(t) < 0$ for all $t$.
$\int_{0}^{a} e^{x-[x]} d x=10 e-9$
where $[x]$ is the greatest integer less than or equal to $x$. Then $a$ is equal to:
(where $C$ is constant of integration)
$f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right) \text {, where } x \in\left[0, \frac{\pi}{2}\right] \text {, }$
consider the following two statements :
($I$) $\mathrm{f}$ is increasing in $\left(0, \frac{\pi}{2}\right)$.
($II$) $\mathrm{f}^{\prime}$ is decreasing in $\left(0, \frac{\pi}{2}\right)$.
Between the above two statements,