Question

compare the right-hand and left-hand derivatives to show that the function is not differentiable at the point P. find all points where f is not differentiable. (graph attached)

Answer 1

$f(x)= \left \{ {{ \sqrt{x} } \atop {2x-1}} \right. \\ \\ f'(x)= \left \{ {{ \frac{1}{2\sqrt{x}} } \atop {2}} \right. \\ \\ f'(1)= \left \{ {{ \frac{1}{2}} \atop {2}} \right. \\ \\ f'(1^-) \neq f'(1^+)$

Therefore, the function is not differentiable at x = 1.

$f(x)= \sqrt{x} \\ \\f'(x)= \frac{1}{2 \sqrt{x} } \\ \\f'(0)=\frac{1}{2 \sqrt{0} } = \infty$

Therefore, the function is not differentiable at x = 0.