Question

If $f\left(x\right)=max\{x^4,x^2,\frac{1}{4}\},\forall x\in [0,\infty ),$ then the sum of square of reciprocal of all the values of $x,$ where $f\left(x\right)$ is not differentiable is:

• A. $5$
• B. $3$
• C. $82$
• D. $\frac{5}{4}$

Answer 1

The correct option is A $5$

$max\{x^4,x^2,\frac{1}{4}\}$ is represented through solid line in the below curve.

Clearly, from the above graph, the points of non differentiablity are at the points where $x^4=x^2$ and $x^2=\frac{1}{4}$
$\Rightarrow x=\pm 1,x=\pm \frac{1}{2}$
In $x\in [0,\infty )$ we have only, $x=1$ and $x=\frac{1}{2}$
$\therefore$ Sum of squares of reciprocal of required values $=1^2+2^2=5$