Maximize And Minimize Functions: A Step-By-Step Guide With An Example

Find max/min value

Plug point original function

To find the maximum or minimum value of a function, we need to take the derivative of the function and then set it equal to zero. The critical points we find are the potential maximum and minimum values of the function. We then need to check the sign of the derivative on either side of these critical points to determine whether they correspond to a maximum or minimum.

Example:

Let’s say we want to find the maximum and minimum values of the function f(x) = x^3 – 6x^2 + 9x + 2 on the interval [0,5].

1. Find the first derivative of the function:
f'(x) = 3x^2 – 12x + 9

2. Set f'(x) equal to 0 and solve for x:
3x^2 – 12x + 9 = 0
x = 1 or x = 3

3. Check the sign of the derivative on either side of the critical points:

When x < 1, f'(x) is negative, so f(x) is decreasing. When 1 < x < 3, f'(x) is positive, so f(x) is increasing. When x > 3, f'(x) is negative, so f(x) is decreasing.

Therefore, we can conclude that x = 1 corresponds to a local minimum and x = 3 corresponds to a local maximum on the interval [0,5].

To find the actual maximum and minimum values of the function, we just need to plug these critical points back into the function:

f(1) = 6
f(3) = -10

So the local minimum value of the function is 6 and the local maximum value of the function is -10.

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