Question

The derivative of f is x^3(x+5)(x-1). At which values of x will the graph of f have a relative maximum? A relative minimum?

Answer 1

The graph can have relative minimum or maximum at points 0, - 5, 1. To determine which is which we should find the sign of the derivative of f on corresponding intervals.

1. $f' \in (-\infty ;-5) => f'<0 => f$ decreases

2. $f' \in (-5;0) => f' > 0=> f$ increases

3. $f' \in (0;1) => f'<0=>f$ decreases

4. $f' \in (1; \infty) => f'>0=>f$ increases

That means $x=-5$ is relative minimum, $x=0$ is relative maximum, $x=1$ is relative minimum of f.