If the function f(x)= cl 1 |x| & ,|x| 2 a x^2+2 b, & |x|<2 . is d… - Vidyadip

MCQ

If the function $f(x)=\left\{\begin{array}{cl}\frac{1}{|x|} & ,|x| \geq 2 \\ a x^2+2 b, & |x|<2\end{array}\right.$ is differentiable on $R$, then $48(a+b)$ is equal to___________.

✓
$15$
B
$16$
C
$75$
D
$78$

✓

Answer

Correct option: A.

$15$

a
$f(x)\left\{\begin{array}{c}\frac{1}{\mathrm{x}} ; \mathrm{x} \geq 2 \\ \mathrm{ax}^2+2 \mathrm{~b} ;-2<\mathrm{x}<2 \\ -\frac{1}{\mathrm{x}} ; \mathrm{x} \leq-2\end{array}\right.$

Continuous at $\mathrm{x}=2 \quad \Rightarrow \frac{1}{2}=\frac{\mathrm{a}}{4}+2 \mathrm{~b}$

Continuous at $\mathrm{x}=-2 \quad \Rightarrow \frac{1}{2}=\frac{\mathrm{a}}{4}+2 \mathrm{~b}$

Since, it is differentiable at $\mathrm{x}=2$

$-\frac{1}{x^2}=2 a x$

Differentiable at $x=2 \quad \Rightarrow \frac{-1}{4}=4 a \Rightarrow a=\frac{-1}{16}, b$

$=\frac{3}{8}$

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