Question

If $f\left(x\right)=min\{1,x^2,x^3\}$, then which among the following options is correct

• A. $f\left(x\right)$ is not everywhere continuous.
• B. $f\left(x\right)$ is continuous and differentiable everywhere.
• C. $f\left(x\right)$ is not differentiable at two points.
• D. $f\left(x\right)$ is not differentiable at one point.

Answer 1

The correct option is D $f\left(x\right)$ is not differentiable at one point.


Drawing curves for $y=x^2,x^3,1$ and represnting $min\{1,x^2,x^3\}$ through a solid line, we get $f\left(x\right)=min\{1,x^2,x^3\}=\{\begin{array}{c}{x}^3,x<1\\ 1,x\ge 1\end{array}$

Clearly, $f\left(1^{+}\right)=f\left(1^{-}\right)=f\left(1\right)=1$
Hence, $f\left(x\right)$ is continuous at $x=1$
But clearly, there is sharp corner at $x=1$
So, $f\left(x\right)$ is not differentiable at $x=1$.