If the function g ( x ) = l a e^x , , , , , ,x 0 b x + x, , ,x > … - Vidyadip

MCQ

If the function $g\left( x \right) = \left\{ \begin{array}{l}
a{e^x},\,\,\,\,\,x \le 0\\
b\cos x + x,\,\,x > 0
\end{array} \right.\,\,\,\,\,\,$ is differentiable, then the value of $a^2 + b^2$ is

A
$5$
✓
$2$
C
$1$
D
$13$

✓

Answer

Correct option: B.

$2$

b
$\mathop {\lim }\limits_{x \to {0^ + }} f\left( x \right) = \mathop {\lim }\limits_{x \to {0^ - }} f\left( x \right) = f\left( 0 \right)$

$\Rightarrow \quad b \quad=a \quad=a$ .......$(1)$

$\mathop {\lim }\limits_{h \to {0^ + }} \frac{{f(0 + h) - f(0)}}{h} = \mathop {\lim }\limits_{h \to {0^ - }} \frac{{f(0 - h) - f(0)}}{h}$

$1=a$ .......$(2)$

$\therefore a=1=b$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from 5. continuity and differentiation→5. continuity and differentiation — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If lines $\frac{x-1}{-3}=\frac{y-2}{2 k}=\frac{z-3}{2}$ and $\frac{x-1}{3 k}=\frac{y-5}{1}$ $=\frac{z-6}{-5}$ are mutually perpendicular, then $k$ is equal to

The function $f ( x )= xe x ^{ x (1- x )}, x \in R$, is

The domain of the function $f(x)=\sin ^{-1}\left(\frac{x^{2}-3 x+2}{x^{2}+2 x+7}\right)$ is.

The function $\text{f(x)}=\begin{cases}\frac{\sin3\text{x}}{\text{x}},&\text{x}\ne0\\\frac{\text{k}}{2},&\text{x}=0\end{cases}$ is continuous at x = 0, then k =

3
6
9
12

The value of $\sin {\cot ^{ - 1}}\tan {\cos ^{ - 1}}x$ is equal to

Let $R$ be the set of real numbers and $f: R \rightarrow R$ be defined by $f(x)=\frac{\{x\}}{1+[x]^2}$, where $[x]$ is the greatest integer less than or equal to $x$, and $\left\{x{\}}=x-[x]\right.$. Which of the following statements are true?

$I.$ The range of $f$ is a closed interval.

$II.$ $f$ is continuous on $R$.

$III.$ $f$ is one-one on $R$

If $y = 2x + {\cot ^{ - 1}}\,x + \log \left( {\sqrt {1 + {x^2}} - x} \right),$ then $y$

The direction cosines of a line which is equally inclined to axes, is given by:

$\underline{+}\frac{1}{3}$
$\underline{+}\frac{1}{\sqrt{3}}$
$1$
$0$

If A = {1, 2, 3}, B = {1, 4, 6, 9} and R is a relation from A to B defined by 'x is greater than y'. The range of R is:

{1, 4, 6, 9}
{4, 6, 9}
{1}
None of these.

Let $f (x) =$ $\left| {\begin{array}{*{20}{c}}{1\, + \,{{\sin }^2}x}&{{{\cos }^2}x}&{4\,\sin \,2x}\\{{{\sin }^2}x}&{1\, + \,{{\cos }^2}x}&{4\,\sin \,2x}\\{{{\sin }^2}x}&{{{\cos }^2}x}&{1\, + \,4\,\sin \,2x}\end{array}} \right|$, then the maximum value of $f (x) =$