Answer:
relative maximum: x = 1
relative minimum: x = 7
Step-by-step explanation:
Critical points:
Values of x for which f'(x) = 0.
Second derivative test:
For a critical point, if f''(x) > 0, the critical point is a relative minimum.
Otherwise, if f''(x) < 0, the critical point is a relative maximum.
Solving a quadratic equation:
Given a second order polynomial expressed by the following equation:
.
This polynomial has roots such that
, given by the following formulas:
In this question:
Finding the critical points:
Simplifying by 3
So
Second derivative test:
The critical points are x = 1 and x = 7.
The second derivative is:
Since f''(1) < 0, at x = 1 there is a relative maximum.
Since f''(x) > 0, at x = 7 there is a relative minumum.