How do I find local maxima and minima of a function?

Question

Marvin Snyder · Accepted Answer

Be that as it may, let f(x) be a real function of a real variable defined in (a,b) and differentiable in the point

x_{0} \in (a, b)

A necessary condition for

x_{0}

to be a local minimum or maximum is that:

f^{'} (x_{0}) = 0

If f(x) is differentiable in the entire interval, or at least in an interval around

x_{0}

, we also have a sufficient condition:

x_{0}

is a local minimum if

f^{'} (x_{0}) = 0

and there is a number

δ

such that:

x \in (x_{0} - δ, x_{0}) \Rightarrow f^{'} (x) \leq 0

x \in (x_{0}, x_{0} + δ) \Rightarrow f^{'} (x) \geq 0

In this case in fact f(x) will be decreasing on the left of

x_{0}

and increasing on the right, so that

x_{0}

is a relative minimum.
Vice versa

x_{0}

is a local maximum if:

x \in (x_{0} - δ, x_{0}) \Rightarrow f^{'} (x) \geq 0

x \in (x_{0}, x_{0} + δ) \Rightarrow f^{'} (x) \leq 0

In both cases the necessary condition is that the derivative of f(x) changes sign around

x_{0}

.
If f(x) also has a second derivative in an interval around

x_{0}

this is equivalent to the conditions:

f^{'} (x_{0}) = 0 and f^{″} (x_{0}) > 0 \Rightarrow x_{0}

is a local minimum.

f^{'} (x_{0}) = 0 and f^{″} (x_{0}) < 0 \Rightarrow x_{0}

is a local maximum.
To determine local maxima and minima, follow these steps:
1) Find the equation's solutions:

f^{'} (x) = 0

also called critical points.
2) Determine the inequality:

f^{'} (x) \leq 0

to see if the sign of f'(x) changes around the critical points, or, alternatively:
2') Calculate f''(x) and look at its value in the critical points.
Example:
Let

f (x) = 2 x^{3} - 3 x^{2} - 12 x + 1

Calculate the derivative:

f^{'} (x) = 6 x^{2} - 6 x - 12

Solve the equation:

f^{'} (x) = 0

6 x^{2} - 6 x - 12 = 0

x^{2} - x - 2 = 0

x = \frac{1 \pm \sqrt{1 + 8}}{2}

so the critical points are:

x_{1} = - 1

x_{2} = 2

As f'(x) is a second degree polynomial with positive leading coefficients we know that:

f^{'} (x) < 0

for

x \in (- 1, 2)

f^{'} (x) > 0

for

x \in (- \infty, - 1) \cup (2, + \infty)

and we can soon determine that

x_{1}

is a local maximum and

x_{2}

a local minimum.
Alternatively we can evaluate the second derivative:

f^{″} (x) = 12 x - 6

and check that:

f^{″} (x_{1}) = - 12 - 6 = - 18 < 0 \Rightarrow x_{1}

is a maximum

f^{″} (x_{2}) = 24 - 6 = 18 > 0 \Rightarrow x_{2}

is a minimum

How do I find local maxima and minima of a function?

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