Question

Let f: R → R be defined by f(x) = x^2· |x|
The derivative f′(x) is:
(A) 3x^2 except at x = 0, where it is undefined.
(B) 3x^2
(C) 3x · |x| except at x = 0, where it is undefined.
(D) 3x · |x|
(E) none of the other choices is correct

Answer 1

\(f(x) = x^{2}\cdot |x|\) is Even and Continuous

One might THINK it is non-differentiable at x = 0, but we have reflected at the point where f'(x) was already zero (although neither minimum nor maximum.)

Answer 2

Let's write f as a piecewise function.
You should know that if x>0 then |x|=x
and if x<0 then |x|=-x
and if x=0 then |x|=0

So we can write your function as