Question

Consider the function f(x) = $\left|x^3\right|$ , where x is real, then the function f(x) at x = 0 is

• A. continuous but not differentiable
• B. once differentiable but not twice
• C. twice differentiable but not thrice
• D. thrice differentiable

Answer 1

The correct option is C twice differentiable but not thrice

f(x) = ${\left|x\right|}^3$ = $\{\begin{array}{ccc}{x}^3& if& x>0\\ -x^3& if& x<0\\ 0& if& x=0\end{array}$

LHL = RHL = f(0) = 0

f(x) is continuous at x = 0

Now, f ' (x) = $\{\begin{array}{ccc}3x^2& if& x>0\\ -3x^2& if& x<0\\ 0& if& x=0\end{array}$
LHD= 0 = RHD
so f(x) is once differentiable

Again f ''(x) = $\{\begin{array}{ccc}6x& if& x>0\\ -6x& if& x<0\\ 0& if& x=0\end{array}$
LHD = 0 = RHD

So f(x) is twice differentiable

Again f '''(x) = $\{\begin{array}{ccc}6& if& x>0\\ -6& if& x=0\end{array}$

LHD = -6 & RHD = 6
LHD $\ne$ RHD

So f(x) is not thrice differentiable.