Lecture #19: Optimization and Applications
MATH 145 – Calculus I
November 3, 2025
Global Maximum and Global Minimum
Last lecture, we discussed how to find local (or relative) extrema.
But what if we want the global extrema (or absolute extrema) of a function?
This is usually called optimizing a function.
Global Maximum and Global Minimum
If \(f(p) \ge f(x)\) for all \(x\) in the domain of \(f\), the global maximum of \(f\) occurs at \(p\).
If \(f(p) \le f(x)\) for all \(x\) in the domain of \(f\), the global minimum of \(f\) occurs at \(p\).
Warning: Whether or not either of these (or both) exists depends of the domain of the function.
The Extreme Value Theorem
If \(f(x)\) is continuous on a closed and bounded interval, then \(f(x)\) has a global maximum and a global minimum.
The question, of course, is where do these extrema occur?
Optimization
Theorem: If \(f(x)\) is a continuous function on an interval \(a \le x \le b\), then the global extrema occur at either:
- The endpoints \(a\) or \(b\).
- The critical points of \(f(x)\) inside the interval \(a < x < b\).
Therefore, to find the global extrema of a function \(f(x)\), you only need to check the endpoints of the interval and the critical points within the interval.
Examples: Optimization
Example: Find the global maximum and minimum of the function \(f(x) = x^4 - 8x^2\) over the interval \([-3, 1]\).
Example: Find the global maximum and minimum of the function \(g(t) = \sin(t) + \cos(t)\) over the interval \(\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2} \right]\).
Example: Find the global maximum and minimum of the function \(h(x) = x e^{-x}\) over the interval \(\left[0, 3 \right]\).
Rule for Extrema on Open Intervals
Theorem: If \(f(x)\) is continuous on an open interval and:
- \(f(x)\) has critical point,
- \(f'(x)\) changes sign at the critical point,
then that critical point is a global extrema – with type determined by the first derivative test.
Example: If possible, find the global maximum and minimum of the function \(f(x) = x + \dfrac{1}{x}\) for \(x > 0\).
Example: Constrained Optimization
Example: Find the dimensions for a closed rectangular box with a square base which has minimal surface area and a fixed volume of \(8\, cm^3\).
Example: A rectangle is inscribed with its base on the \(x\)-axis and its upper corners on the parabola \(y = 1 - x^2\). What are the dimensions of such a rectangle with the greatest possible area?
Example: A rectangular box with a square bottom and closed top is to be made from two materials. The material for the sides costs $1.50 per square foot and the material for the top and bottom costs $3.00 per square foot. If you are willing to spend $15 on the box, what is the largest volume it can contain?
