Local Maxima and Minima, and, Absolute Maxima and Minima

# Local Maxima and Minima

 Definition: Suppose that $f$ is a function and $D(f)$ is the domain of $f$. Let $a \in D(f)$. We say that $f(a)$ is a Local (Relative) Maximum Value on $f$ or a Local (Relative) Maxima if $f(a) ≥ f(x)$ when $x$ is near $a$. Similarly, we say that $f(a)$ is a Local (Relative) Minimum Value on $f$ or a Local (Relative) Minima if $f(a) ≤ f(x)$ when $x$ is near $a$. The term "near $a$" means that there exists an open interval centered at $a$ for which a certain property holds. That is, $f$ has a local maximum at $a$ with value $f(a)$ if there is an open interval $(c, d)$ centered at $a$ such that $f(x) \leq f(a)$ for all $x \in (c, d)$.

The term Extrema is used to denote maximum and minimum values.

Fortunately, it is relatively easy to determine the coordinates of local minima and maxima because the tangent lines at those coordinates have a slope of zero, and thus, if $f$ is a differentiable function, then if $f'(x) = 0$, the point $(x, f(x))$ is a candidate point for a local maximum or minimum of $f$.

Note that in general $f'(c) = 0$ does NOT imply that $f(c)$ is a local maximum or local minimum value. For example, consider the function $f(x) = x^3$. Then $f'(x) = 3x^2$. So $f'(0) = 0$, but $(0, 0)$ is clearly not a local maximum or local minimum to $f$.

## Example 1

Find all local maxima and minima of the function $f(x) = x^3 - 2x + 1$.

We first differentiate this function to get $f'(x) = 3x^2 - 2$, and then set $f'(x) = 0$ to solve for $x$:

(1)
\begin{align} 0 = 3x^2 - 2 \\ 3x^2 = 2 \\ x^2 = \frac{2}{3}\\ x = \pm \sqrt{\frac{2}{3}} \approx \pm 0.816... \end{align} Graphically, we can see that a local maxima and a local minima exist at $x = -0.816$ and $x = 0.816$. We plug these values into $f$ to get the coordinates as $(0.816..., -0.08...)$ and $(-0.816..., 2.08...)$.

# Absolute Maxima and Minima

 Definition: Suppose that $f$ is a function and $D(f)$ is the domain of $f$. Let $a \in D(f)$. We say that $f(a)$ is the Absolute (Global) Maximum on $f$ or the Absolute (Global) Maxima if $f(a) ≥ f(x)$ for all other $x \in D(f)$. Similarly, we say that $f(a)$ is the Absolute (Global) Minimum on $f$ or the Absolute (Global) Minima if $f(a) ≤ f(x)$ for all other $x \in D(f)$. Note: From our definition of absolute maxima and minima, if $(a, f(a))$ is an absolute max/min, then it is also a local max/min too.

## Example 2

Show that $f(x) = \sin x$ has infinitely many absolute maxima and absolute minima on the interval $(-\infty, \infty)$.

When we differentiate $f$ we get $f'(x) = \cos x$, and when we set $f'(x) = 0 = \cos x$, we get $x = ..., -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}, ...$. We note that for $k \in \mathbb{Z}$, $\: f(\frac{\pi}{2} + 2k\pi) = 1$, and $f(\frac{-\pi}{2} + 2k\pi) = -1$. Since $R(f) = [-1, 1]$, there are infinitely many absolute maxima and absolute minima.