Stationary points of a given function

Question

If given f(x) = x³e³,

a) How can I find coordinates of the stationary point(s) of f(x)

b) How can I determine the nature of the stationary point of f(x).

c) On what interval is the function f(x) increasing?

Answer 1

a) f'(x) = 3x²e³ = 0; x² = 0; x = 0; f(0) = 0. So, (0, 0) is stationary point of f(x)

b) Because f'(x) do not change sign around x = 0 at x = 0 f(x) has no extrima.

c) Because f'(x) = 3x²e³ ≥ 0 for all real x, f(x) increasing on innterval (- ∞, ∞)