Question

If f(x)=|x|3, show that f"(x)exists for all real x, and find it.

Answer 1

It is known that,|x|=\left\\{\begin{matrix} x,&if\,x\geq 0 \\\ -x,&if\,x<0 \end{matrix}\right.

if x<0 Therefore, when x≥0,

f(x)=|x|3=x3 In this case,

f'(x)=3x2 and hence,f""(x)=6x

when x<0,f(x)=|x|3=(-x)3=-x3 In this case,f'(x)=-3x2 and hence,f""(x)=-6x Thus,for f(x)=|x|3

f""(x)exists for all real x and is given by f'(x)=\left\\{\begin{matrix} 6x,&if\,x\geq 0 \\\ -6x,&if\,x<0 \end{matrix}\right.