Applications of derivatives · Definition

Optimization problems

calculusapplications

Before you start you need to know

Here a quantity is modeled as a function of one variable and its extreme values are sought on an allowed interval.

Definition

An optimization problem seeks the largest or smallest value of a quantity subject to constraints, expressed as a function of one variable.

\[\text{maximize }A(x)\ \Rightarrow\ A'(x)=0\]

Optimizing means finding the best possible value of something (maximum area, minimum cost) under a constraint. The real work is modeling: choosing variables and writing the quantity to optimize as a function of a single variable.

Then you apply what you already know: differentiate, find critical points and compare with the endpoints of the allowed interval. Always check the answer makes sense (positive lengths, domain).

If a continuous function is defined on a closed bounded interval, the extreme value theorem ensures it attains a maximum and a minimum.

Think about it

With 40 m of fence, which rectangle has maximum area?

The 10 m by 10 m square: A = x(20−x) is maximized at x = 10.

Exercises 10

  1. 80 m of fencing is used to enclose a rectangular plot. What is the maximum possible area, in m²?
  2. 120 m of fencing is used to enclose a rectangular plot. What is the maximum possible area, in m²?
  3. Two positive numbers have product 36. What is the smallest possible value of their sum?
  4. Equal squares are cut from the corners of a 12 cm square sheet and the sheet is folded into an open box. What is the maximum volume, in cm³?
  5. What is the maximum value of \(f(x)=-x^{2}+8x+5\)?
  6. A rectangle has perimeter 48 m. What is the maximum possible area, in m²?
  7. Two positive numbers add up to 20. What is the largest possible value of their product?
  8. What is the minimum value of \(f(x)=x^{2}-6x+13\)?
  9. What is a critical point of a function \(f\)?
  10. Maximize \(A(x)=x(10-x)\) on the interval \([0,4]\). What is the maximum value?

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.