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# 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