# 4.2 | Optimization

The Extreme Value Theorem:
If $$f$$ is continuous on the closed interval $$a \leq x \leq b,$$ then $$f$$ has a global maximum and a global minimum on that interval.
Global Maxima and Minima on a Closed Interval: Test the Candidates
For a continuous function $$f$$ on a closed interval $$a \leq x \leq b:$$

• Find the critical points of $$f$$ in the interval.
• Evaluate the function at the critical points and at the endpoints, $$a$$ and $$b.$$ The largest value of the function is the global maximum; the smallest value of is the global minimum.
Global Maxima and Minima on a Open Interval or All Real Numbers
For a continuous function $$f,$$ find the value of $$f$$ at all the critical points and sketch a graph. Look at values of $$f$$ when $$x$$ approaches the endpoints of the interval, or approaches $$\pm \infty,$$ as appropriate. If there is only one critical point, look at the sign of $$f’$$ on either side of the critical point.