Question

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

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

Answer 1

The correct option is C Both 1 and 2

Statement 1

$f\left(x\right)=xforx\ge 0$

$f\left(x\right)=-xforx<0$

For a function to be differentiable at a point, it's derivative must be continuous at that point

$\Rightarrow$ Left hand derivative(L.H.D)=Right hand derivative (R.H.D)

$L.H.D=f^{{}^{\prime }}\left(x\right)forx<0$

$\Rightarrow L.H.D=-1$

$R.H.D=f^{{}^{\prime }}\left(x\right)forx\ge 0$

$\Rightarrow R.H.D=1$

$\Rightarrow L..H.D\ne R.H.D$

The function $f\left(x\right)=\left|x\right|$ is not differentiable at $x=0$

Statement 2

Derivative of function $f\left(x\right)=e^x,f^{{}^{\prime }}\left(x\right)=e^x$

$f^{{}^{\prime }}\left(x\right)isalwayscontinuousforallx\in R$

$\Rightarrow$ The function $f\left(x\right)=e^x$ is differentiable at $x=1$

Both statement 1 and 2 are correct.