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If f is differentiable at a, a is a local extreme if:

## 1.

f'(a) = 0. The best Maths tutors available
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1st lesson free!  4.9 (23 reviews)
Intasar
£42
/h
1st lesson free!  5 (17 reviews)
Matthew
£25
/h
1st lesson free!  4.9 (6 reviews)
Dr. Kritaphat
£49
/h
1st lesson free!  4.9 (11 reviews)
Paolo
£25
/h
1st lesson free!  5 (28 reviews)
Ayush
£60
/h
1st lesson free!  4.9 (9 reviews)
Petar
£27
/h
1st lesson free!  4.9 (11 reviews)
Rajan
£15
/h
1st lesson free!  5 (13 reviews)
Farooq
£35
/h

f''(a) ≠ 0.

## Local Maxima

If f and f' are differentiable at a, a is a local maximum if:

f'(a) = 0

f''(a) < 0

## Local Minima

If f and f' are differentiable at a, a is a local minimum if:

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

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