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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
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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