If the function f(x) = { {1/|x|, |x| ≥ 3}, {ax^2 + 2b, |x| < 2} is differentiable on R, then

Question

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

Answer 1

Correct answer: 15

\(f(x)=\left\{\begin{array}{cc}\frac{1}{x}; & x \geq 2 \\ a x^{2}+2 b; & -2<x<2\\-\frac1x; &x \le -2\end{array}\right.\)

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 \(\mathrm{x}=2\)

\(-\frac{1}{x^{2}}=2 \mathrm{ax}\)

Differentiable at \(\mathrm{x}=2 \)

\(\Rightarrow \frac{-1}{4}=4 \mathrm{a} \Rightarrow \mathrm{a}=\frac{-1}{16}, \mathrm{~b}=\frac{3}{8}\)

\(48(a + b) = 48\left(\frac{-1}{16} + \frac 38 \right) = 15\)