Limits, continuity and differentiability — practice questions

Question 1 · difficulty L2 · understanding

Let $f(x)$ and $g(x)$ be twice differentiable functions satisfying $f^{\prime \prime}(x)=g^{\prime \prime}(x)$ for all $x \in \mathbf{R}, f^{\prime}(1)=2 g^{\prime}(1)=4$ and $g(2)=3 f(2)=9$. Then $f(25)-g(25)$ is equal to :

• A. 20
• B. 40
• C. -20
• D. -40

Question 2 · difficulty L3 · understanding

Let $f(x)=\left\{$$\begin{array}{ll}x^3+8 ; & x<0, \\ x^2-4 ; & x \geq 0,\end{array}$$\right.$and$g(x)= $$\begin{cases}(x-8)^{1 / 3} ; & x<0, \\ (x+4)^{1 / 2} ; & x \geq 0 .\end{cases}$$$Then the number of points, where the function$g \circ f$is discontinuous, is$\_\_\_\_$ .

Question 3 · difficulty L3 · understanding

Let $f(x)=\left\{$$\begin{array}{cc}e^{x-1} & , x<0 \\ x^2-5 x+6 & , x \geq 0\end{array}$$\right.$and$g(x)=f(|x|)+|f(x)|$. If the number of points where$g$is not continuous and is not differentiable are$\alpha$and$\beta$respectively, then$\alpha+\beta$is equal to$\_\_\_\_$

Question 4 · difficulty L3 · understanding

The number of points, at which the function $f(x)=\max \left\{6 x, 2+3 x^2\right\}+|x-1| \cos \left|x^2-\frac{1}{4}\right|, x \in(-\pi, \pi)$, is not differentiable, is $\_\_\_\_$ .

Question 5 · difficulty L3 · understanding

The number of points in the interval $[2, 4]$, at which the function $f(x) = \left[ x^2 - x - \frac{1}{2} \right]$, where $[ \cdot ]$ denotes the greatest integer function, is discontinuous, is ________.

Question 6 · difficulty L3 · understanding

Let $m$ and $n$ be the number of points at which the function $f(x)=\max \left\{x, x^3, x^5, \ldots x^{21}\right\}, x \in \mathbb{R}$, is not differentiable and not continuous, respectively. Then $m+n$ is equal to _________.

Question 7 · difficulty L3 · understanding

If\,\,\mathop {\lim }\limits_{x \to 0} \left(\frac{\tan x}{x}\right)^{\frac{1}{x^2}}=p \text {, then } 96 \log _{\mathrm{e}} p \text { is equal to____________ }

Question 8 · difficulty L3 · understanding

Let $\mathrm{f}(x)=\left\{$$\begin{array}{lc}3 x, & x<0 \\ \min \{1+x+[x], x+2[x]\}, & 0 \leq x \leq 2 \\ 5, & x>2\end{array}$$\right.$where [.] denotes greatest integer function. If$\alpha$and$\beta$are the number of points, where$f$is not continuous and is not differentiable, respectively, then$\alpha+\beta$ equals _______ .

Question 9 · difficulty L3 · understanding

Let the function, $$f(x)= \begin{cases}-3 \mathrm{ax}^2-2, & x<1 \\ \mathrm{a}^2+\mathrm{b} x, & x \geqslant 1\end{cases}$$ be differentiable for all $x \in \mathbf{R}$, where $\mathrm{a}>1, \mathrm{~b} \in \mathbf{R}$. If the area of the region enclosed by $y=f(x)$ and the line $y=-20$ is $\alpha+\beta \sqrt{3}, \alpha, \beta \in Z$, then the value of $\alpha+\beta$ is ___________ .

Question 10 · difficulty L3 · understanding

The value of \lim _\limits{x \rightarrow 0} 2\left(\frac{1-\cos x \sqrt{\cos 2 x} \sqrt[3]{\cos 3 x} \ldots \ldots . \sqrt[10]{\cos 10 x}}{x^2}\right) is __________.