Question

Let $f\left(x\right)=\{\begin{array}{cc}4x^2+1,& x<\frac{1}{2}\\ 2(\ln x+1),& \frac{1}{2}\le x<1\\ x^2+1,& x\ge 1\end{array}$
then which of the following is/are true?

• A. $f\left(x\right)$ has two points of extremum
• B. $f\left(x\right)$ is continuous $\forall xϵR,$ except at one point
• C. $f\left(x\right)$ is differentiable $\forall xϵR,$ except at one point
• D. $f\left(x\right)$ has exactly one point of minima

Answer 1

Answer: (A), (B), (C)

$f\left(x\right)$ is differentiable $\forall xϵR,$ except at one point


So non-differentiable at one point, discontinuous at one point and have two points of local minima at $x=0\text{and}x=\frac{1}{2}.$