Question

If the function $f\left(x\right)=|x^2+a|x|+b|$ has exactly three points of non-differentiability, then which of the following may hold?

• A. b = 0, a < 0
• B. $b<0,a\in R$
• C. $b>0,a\in R$
• D. $b<0,a\in R^{-}$

Answer 1

The correct option is A

b = 0, a < 0

b = 0, a < 0

$f\left(x\right)=x^2+ax+b$ is quadratic polynomial

cut x axis at x = 0 and x = – a

$f\left(x\right)=x^2+ax+b$

$f\left(|x|\right)=x^2+a|x|+b$

$|f|x||=|x^2+a|x|+b|$

Exactly at three points function is not differentiable.