Maxima and Minima

If f is differentiable at a, a is a local extreme if:

1. f'(a) = 0.

2. f''(a) ≠ 0.

Local Maxima

If f and f' are differentiable at a, a is a local maximum if:

1. f'(a) = 0

2. f''(a) < 0

Local Minima

If f and f' are differentiable at a, a is a local minimum if:

1. f'(a) = 0

2. f''(a) > 0

Calculation of the Maximum and Minimum

Study the maximum and minimum of the following function:

f(x) = x3 − 3x + 2

To find the local extremes, follow these steps:

1. Calculate the first derivative and its roots.

f'(x) = 3x2 − 3 = 0

x = −1 x = 1.

2. Calculate the 2nd derivative, and determine the sign that the zeros take from the first derivative:

f''(x) > 0 Minimum.

f''(x) < 0 Maximum.

f''(x) = 6x

f''(−1) = −6 Maximum.

f'' (1) = 6 Minimum.

3. Calculate the image (in the function) of the relative extremes.

f(−1) = (−1)3 − 3(−1) + 2 = 4

f(1) = (1)3 − 3(1) + 2 = 0

Maximum (−1, 4) Minimum (1, 0)


If the increase and decrease of a function has been studied the following can be determined:

1. The maximum points of the function, in which it passes from increasing to decreasing.

2. The minimum points of the function, in which it passes from decreasing to increasing.


Example

Find the maximum and minimum:

Function

Maximum and Minimum Operations

Maximum and Minimum Operations

Function Roots

Maximum and Minimum Operations

There is a minimum at x = 3.

Maximum and Minimum Solutionminimum(3, 27/4)

At x = 1, there is no maximum for x = 1 because it does not belong in the domain of the function.