Ximpledu

Local Maximum, Minimum

See how to find the local maximum, minimum
(+ global maximum, minimum).
3 examples and their solutions.

Increase, Decrease

Increase


f'(x): (+)
If f'(x) is (+),
then the graph of y = f(x) increases. (↗)
f'(x): Slope of the curve

Decrease


f'(x): (-)
If f'(x) is (-),
then the graph of y = f(x) decreases. (↘)

Local Maximum, Minimum

Local Maximum


f'(x): (+)  (-)
A local maximum is the point
where f'(x) changes from (+) to (-).
If f(x) is differentiable,
the local maximum is the point that satisfies
f'(x): (+)0(-)
(Left graph)
If f(x) is not differentiable,
the local maximum is the point that satisfies
f'(x): (+)  (-).
(Right graph)

Local Minimum


f'(x): (-)  (+)
A local minimum is the point
where f'(x) changes from (-) to (+).
If f(x) is differentiable,
the local minimum is the point that satisfies
f'(x): (-)0(+)
(Left graph)
If f(x) is not differentiable,
the local minimum is the point that satisfies
f'(x): (-)  (+).
(Right graph)

Example

f(x) = x3 - 3x2 - 9x + 7
Local maximum, minimum?
Solution

Example

f(x) = x4 - 4x3 + 10
Local maximum, minimum?
Solution

Global Maximum, Minimum

Global Maximum

The greatest value of the graph.

Global Minimum

The least value of the graph.

Example

f(x) = x3 - 3x + 1 [0, 2]
Global maximum, minimum?
[0, 2]: 0 ≤ x ≤ 2

Solution