Question

How do you find the exact relative maximum and minimum of the polynomial function of `f(x) = x^3 - 3x^2 - x + 1`?

Answer 1

The first derivative finds the inflection points. The second derivative indicates whether it is a maximum or minimum.

Explanation:

Suppose f(x) is a function of x that is twice differentiable at a stationary point x_0.
1. If `f''(x_0)>0`, then f has a local minimum at x_0.
2. If `f''(x_0)<0`, then f has a local maximum at x_0.

Weisstein, Eric W. "First Derivative Test." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/FirstDerivativeTest.html
Weisstein, Eric W. "Second Derivative Test." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SecondDerivativeTest.html

Inflection points at:
`f(x) = x^3 − 3x^2 − x + 1`
`f'(x) = 3x^2 − 6x -1 = 0`

Local min/max from:
`f''(x) = 6x − 6` at the inflection points.