Continuity And Differentiability Question 6
Question 6 - 2024 (30 Jan Shift 1)
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
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Answer (15)
Solution
$f(x)\left{\begin{array}{c}\text { athon } \frac{1}{x} ; \mathrm{x} \geq 2 \ \mathrm{ax}^{2}+2 \mathrm{~b} ;-2<\mathrm{x}<2 \ \text { atho }-\frac{1}{\mathrm{x}} ; \mathrm{x} \leq-2 \text { athor }\end{array}\right.$
Continuous at $\mathrm{x}=2 \Rightarrow \frac{1}{2}=\frac{\mathrm{a}}{4}+2 \mathrm{~b}$
Continuous at $\mathrm{x}=-2 \Rightarrow \frac{1}{2}=\frac{\mathrm{a}}{4}+2 \mathrm{~b}$
Since, it is differentiable at $\mathrm{x}=2$
$-\frac{1}{x^{2}}=2 a x$
Differentiable at $x=2 \Rightarrow \frac{-1}{4}=4 a \Rightarrow a=\frac{-1}{16}, b=\frac{3}{8}$
