Question

How do you find the exact relative maximum and minimum of the polynomial function of `f(x) = x^3-6x^2+15`?

Answer 1

Relative maximum: `(0,15)`
Relative minimum: `(4,-17)`

Explanation:

Relative maximums and minimums occur whenever the derivative equals 0. Using the power rule, we find the derivative is:
`f'(x)=3x^2-12x`

Setting it equal to `0` yields:
`0=3x^2-12x`
`0=x^2-4x`
`0=x(x-4)`
`x=0` and `x=4`

To find the exact minimum and maximum, we evaluate these two `x`-values (called critical points):
`f(0)=(0)^3-6(0)^2+15=15->` maximum
`f(4)=(4)^3-6(4)^2+15=64-96+15=-17->` minimum

Looking at the graph of the function, we can see that there is indeed a local max at `(0,15)` and a local min at `(4,-17)`.
graph{x^3-6x^2+15 [-42.95, 49.5, -19.6, 26.67]}