Consider the following statements:

1. The function f(x) = | x | is not differentiable at x = 1

2. The function f(x) = e^x is differentiable at x = 0.

Which of the above statements is/are correct?

Concept:

Differentiability of a function:

We will define the differentiability of the function with the help of its graph. If a graph of a function is smooth everywhere then the function is said to be differentiable everywhere in the xy - plane.

If the function has a sharp edge or a vertical asymptote (tangent) to the curve then the function is not differentiable at that point.

Calculation:

Observe the graph of the function f(x) = |x|.

At x = 1 the curve is smooth and thus the function is differentiable at x = 1.

On the other hand the function is not differentiable at x = 0 as it has a corner at x = 0.

Observe the graph of the function f(x) = e^x.

At x = 0 the curve is smooth so it is differentiable at x = 0.

Thus only the second statement is true.