Determine Where a Function Is Increasing, Decreasing, or Constant

Absolute Maxima and Minima

The absolute maximum of $f$ at $x=c$ is $f(c)$ where $f(c) \geq f(x)$ for all $x$ in the domain of $f$. The absolute minimum of $f$ at $x=d$ is $f(d)$ where $f(d) \leq f(x)$ for all $x$ in the domain of $f$.

Example 10

Finding Absolute Maxima and Minima from a Graph

For the function $f$ shown in Figure 14, find all absolute maxima and minima.

Figure 14

Solution

Observe the graph of $f$. The graph attains an absolute maximum in two locations, $x=-2$ and $x=2$, because at these locations, the graph attains its highest point on the domain of the function. The absolute maximum is the $y$-coordinate at $x=-2$ and $x=2$, which is 16.

The graph attains an absolute minimum at $x=3$, because it is the lowest point on the domain of the function's graph. The absolute minimum is the $y$-coordinate at $x=3$, which is $-10$.