Examples of using the derivative tests for finding maximum/minimum

by Kurious Fox

  • Problem 1: Find the maximum value of the function f(x) = -2x^2 + 4x + 1 .
  • Solution:
    • First, find the derivative f'(x) = -4x + 4 .
    • Set the derivative equal to zero to find critical points:
      -4x + 4 = 0 ? x = 1 .
    • To determine if this critical point is a maximum or minimum, find the second derivative f''(x) = -4 .
    • Since f''(x) < 0 , the function has a maximum at x = 1 .
    • The maximum value is f(1) = -2(1)^2 + 4(1) + 1 = 3 .

Problem 2: Find the minimum distance from the point (2, 3) to the line y = x + 1 .

Solution:

  1. Let the point on the line be (x, x + 1) .
  2. The distance D from (2, 3) to (x, x + 1) is given by:
    D = \sqrt{(x - 2)^2 + ((x + 1) - 3)^2}
  3. Simplify the distance formula:
    D = \sqrt{(x - 2)^2 + (x - 2)^2} = \sqrt{2(x - 2)^2} = \sqrt{2}|x - 2|
  4. To minimize D , minimize |x - 2| . The minimum value of |x - 2| is 0, which occurs at x = 2 .
  5. Substitute x = 2 back into the distance formula:
    D = \sqrt{2}(2 - 2) = 0
  6. Check if x = 2 is a minimum using derivatives:
    D^2 = 2(x - 2)^2
  7. Differentiate D^2 with respect to x :
    \frac{d(D^2)}{dx} = 4(x - 2)
    Set the derivative to 0:
    4(x - 2) = 0 \implies x = 2
  8. The second derivative test:
    \frac{d^2(D^2)}{dx^2} = 4
    Since \frac{d^2(D^2)}{dx^2} > 0 , x = 2 is indeed a minimum.

More examples:

Example of using derivatives to optimize the material cost
Example of using derivatives to find optimal drug dosage

Discover more from Science Comics

Subscribe to get the latest posts sent to your email.

Related Posts

Time machine? How can scientists predict the population of owls in the future?

This funny comic illustrates how scientists forecast future animal population sizes using various methods, including mathematical models, data collection through field surveys, and statistical techniques to analyze trends. They incorporate environmental factors and utilize simulation software to predict changes. These approaches are essential for effective conservation and wildlife management strategies.

Leave a Reply

error: Content is protected !!