Question

If $f\left(x\right)=\{\begin{array}{c}a{x}^2-bif\left|x\right|<1\\ -\frac{1}{\left|x\right|}if\left|x\right|\ge 1\end{array}$ is derivable at $x=1$ then the values of $a+b$ is

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The correct option is C 2

for $f\left(x\right)$ to be derivable ,$f\left(x\right)$ must be continuous

$⟹a-b=-1$

When $\mid x\mid <1,f^{\prime }\left(x\right)=2ax$

When $\mid x\mid \ge 1,f^{\prime }\left(x\right)=\frac{\mid x\mid }{x^3}$

at $x=1$ $2a=1⟹a=\frac{1}{2}⟹b=\frac{3}{2}$

$a+b=2$