To find maximum and minimum values such as area and volume, reduce the variables to one, express the quantity as a function, and differentiate.
Example ① Among rectangles with a perimeter of 20cm, find the one with the greatest area.
Solution:Let the width be x and the length be 10−x（0
S' = 10 − 2x = 0 ⇒ x = 5。S(5) = 25。
The maximum area is 25cm² when the rectangle is 5×5 (a square)
Solution:Let the width be x and the length be 10−x（0
The maximum area is 25cm² when the rectangle is 5×5 (a square)
Key Point
- “Reduce the number of variables to one” is the first step
- Do not forget to check the domain
Practice Problems
- Find the maximum area of a rectangle inscribed in a circle of radius 1。
Solutions and Explanations
- Solution:2
The diagonal is 2, the width is 2cosθ, and the length is 2sinθ. S=2sin(2θ), with maximum 2。