Question

Show why the function f(x) = 3rad x is not differentiable at x=0

Answer 1

$f(x) = 3\sqrt{x}$
$f'(x) = -\frac{3}{2\sqrt{x}}$

When x = 0, f'(0) becomes undefined. Thus, we know that there is no derivative at x = 0. You could also show it by taking the limits, which is the fundamental of calculus. Show that as x approaches 0 from the negative side is not equal to the value as x approaches 0 from the positive side.

What this tells us is that there is no uniquely distinct point that satisfies the continuity of the curve in order for the graph to have a specific gradient at x = 0; thus, this means there is no gradient at x = 0, and hence, the function is not differentiable at x = 0