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:
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.
