The total number of points of non-differentiability of $f\left(x\right)=\max \left\{{\sin }^{2}x,{\cos }^{2}x,\frac{3}{4}\right\}$ in $[0,10\pi ]$, is

 Here, $f\left(x\right)=\max \left\{{\sin }^{2}x,{\cos }^{2}x,\frac{3}{4}\right\}$

Thus, representing $f\left(x\right)$ grapghically

As we know that $f\left(x\right)$ is non-differentiable at points where its graph has sharp edge points.

Since, ${\sin }^{2}x$ and ${\cos }^{2}x$ are periodic with period $\pi$ and in $[0,\pi ]$, there are four points of non-differentiability of $f\left(x\right)$ because of sharp edge points.

Thus, in $[0,10\pi ]$, there are $40$ points of non-differentiability.

Hence option-$1$ is the correct answer.