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

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

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

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

$\therefore a=1=b$

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