Consider the following statements:
1. The function `f(x)=|x|` is not differentiable at x=1.
2. The function `f(x)=e^(x)` is not differentiable at x=0.
Which of the above statements is/are correct ?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Answer

B

Explanation

Statement 1: f(x)=|x|

From the graph, the curver has sharp turn at x=0.
Therefore, the function f(x)=|x| is not differentiable only x=0, it is differentiable at x=1
Statement 2 : `f(x)=e^(x)`
`Rf'(0)=underset(hto0)lim(f(0+h)-f(0))/(h)`
`=underset(hto0)lim(e^((0+h))-e^(@))/(h)=underset(hto0)lim(e^(h)-1)/(h)`
Use L' Hospital rule
`=underset(hto0)lim(e^(h)-0)/(1)=e^(@)=1`
`f'(0)=underset(hto0)lim(1-e^(-h))/(h)`
Use L' Hospital rule
`=underset(hto0)lim(e^(-h))/(1)=e^(-0)=1`
Therefore `f(x)=e^(x)` is differentiable at x=1.