Question

What are the extrema of `f(x) = x^3 - 27x`?

Answer 1

`(-3, 54) and (3, -54)`

Explanation:

Relative maximum and minimum points occur when the derivative is zero, that is when `f'(x)=0`.
So in this case, when `3x^2-27=0`
#=>x=+-3.

Since the second derivative `f''(-3)<0 and f''(3)>0`, it implies that a relative maximum occurs at x=-3 and a relative minimum at x=3.

But: `f(-3)=54 and f(3)=-54` which implies that `(-3, 54)` is a relative maximum and `(3, -54)` is a relative minimum.

graph{x^3-27x [-115.9, 121.4, -58.1, 60.5]}