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:

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.

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.