askvity

How do you find the extreme value of a function?

Published in Calculus Optimization 2 mins read

To find the extreme values (local maximums and minimums) of a function, you typically use the derivative. Here's a step-by-step guide:

Steps to Find Extreme Values

  1. Find the Derivative: Calculate the first derivative of the function, denoted as f'(x).

  2. Set the Derivative to Zero: Set the derivative equal to zero: f'(x) = 0.

    • According to the reference, setting f'(x) = 0 allows us to find the x-coordinates of potential extreme values (local maximums and minimums).

  3. Solve for x: Solve the equation f'(x) = 0 for x. The solutions are called critical points.

  4. Determine if it's a Max or Min (or Neither): Use either the first derivative test or the second derivative test to determine the nature of each critical point:

    • First Derivative Test:
      • Choose test values on either side of the critical point.
      • Evaluate f'(x) at these test values.
      • If f'(x) changes from positive to negative at the critical point, it's a local maximum.
      • If f'(x) changes from negative to positive at the critical point, it's a local minimum.
      • If f'(x) doesn't change sign, it's neither a maximum nor a minimum (it's a saddle point or inflection point).
    • Second Derivative Test:
      • Find the second derivative, f''(x).
      • Evaluate f''(x) at each critical point.
      • If f''(x) > 0, the critical point is a local minimum.
      • If f''(x) < 0, the critical point is a local maximum.
      • If f''(x) = 0, the test is inconclusive, and you should use the first derivative test.

  5. Find the y-coordinate: Substitute the x-values of the extreme values into the original function, f(x), to find the corresponding y-values. This gives you the coordinates of the extreme values.

Example

Let's say we have the function f(x) = x3 - 3x.

  1. f'(x) = 3x2 - 3

  2. 3x2 - 3 = 0

  3. 3x2 = 3 => x2 = 1 => x = ±1

  4. Using the Second Derivative Test:
    f''(x) = 6x

    • f''(-1) = -6 < 0, so x = -1 is a local maximum.
    • f''(1) = 6 > 0, so x = 1 is a local minimum.

  5. Find the y-coordinates:

    • f(-1) = (-1)3 - 3(-1) = -1 + 3 = 2. Local maximum at (-1, 2).
    • f(1) = (1)3 - 3(1) = 1 - 3 = -2. Local minimum at (1, -2).

Related Articles