AP Calculus BC / Unit 5: Analytical Applications of Differentiation
5.11 Solving Optimization Problems
Notes
Steps: draw and label, write the objective function, use the constraint to get one variable, differentiate, find critical points, then justify the max or min (first derivative, second derivative, or candidates test).
Practice
A box with a square base and no top has volume 32. Minimize surface area $\displaystyle S=x^2+\frac{128}{x}$. The best $x$ is:
$\displaystyle 2x-\frac{128}{x^2}=0$ gives $x^3=64$.
Two positive numbers sum to 12. Their product is largest when they are:
$x(12-x)$ peaks at $x=6$.
After finding a critical point, the next step is:
A critical point could be either, or neither.
Ready for more? Take the Unit 5 test.