# Maxima and Minima

If f is differentiable at * a*,

*is a*

**a****local extreme**if:

1. **f'(a) = 0**.

2. **f''(a) ≠ 0**.

#### Local Maxima

If f and f' are differentiable at * a*,

*is a*

**a****local maximum**if:

1. **f'(a) = 0**

2. **f''(a) < 0**

#### Local Minima

If f and f' are differentiable at * a*,

*is a local minimum if:*

**a**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) = x^{3} − 3x + 2

To find the local extremes, follow these steps:

1. Calculate the first derivative and its roots.

f'(x) = 3x^{2} − 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:

There is a minimum at x = 3.

minimum(3, 27/4)

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