Define a function f: R R by f(x)= cc x^2 x , & for x<0 x^2+a x+b,… - Vidyadip

MCQ

Define a function $f: R \rightarrow R$ by $f(x)=\left\{\begin{array}{cc}\frac{\sin x^2}{x}, & \text { for } x<0 \\ x^2+a x+b, & \text { for } x \geq 0\end{array}\right.$ Suppose $f(x)$ is differentiable of $R$. Then,

A
$a=0, b=0$
✓
$a=1, b=0$
C
$a=0, b=1$
D
$a=1, b=1$

✓

Answer

Correct option: B.

$a=1, b=0$

b
(b)

We have, $f: R \rightarrow R$ by

$f(x)=\left\{\begin{array}{cc}\frac{\sin x^2}{x}, & x < 0 \\x^2+a x+b, & x \geq 0\end{array}\right.$

$f(x)$ is differentiable on $R$.

$\therefore f(x)$ is also continuous on $R$.

$\begin{aligned}\lim _{x \rightarrow 0^{-}} f(x) &=\lim _{x \rightarrow 0^{+}} f(x) \\ \lim _{x \rightarrow 0^{-}} \frac{\sin x^2}{x} &=\lim _{x \rightarrow 0^{+}} x^2+a x+b \\\Rightarrow \quad &=b \Rightarrow b=0\end{aligned}$

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