How do the tangents,derivatives,continuity,diffrentiability of a function get affected if the function is squared?

Question

Please explain graphically the above changes(I imply changes in roots, derivatives at some points, continuity and differentiability at some points etc

Answer 1

Take the function $f(x) = 1$ if $x$ is rational and $-1$ if $x$ is irrational. Then $f$ is no where differentiable or continuous, and its graph has no tangent lines anywhere.

But $f^2(x) =1 $ for all real $x$, so it's differentiable and continuous everywhere and has a tangent at every point.

Answer 2

If y is f(x) then

$$(y^2)^{'} = 2 y y^{'}$$

The place where max/min occurs (x coordinate) is unaffected. But max / min of $y$ is itself changed as do others mentioned in title line.