Question

the functions $f(x)$ and $g(x)$ are both continuous functions. the function $f(x)$ is defined algebraically and the function $g(x)$ is defined graphically.
use the drop-down menus to compare the maximum of $f(x)$ to the maximum of $g(x)$.

$f(x)=\

$$\begin{cases} \\frac{5}{2}x + 18 & \\text{if } x \\leq -6 \\\\ 3 & \\text{if } -6 < x \\leq -2 \\\\ \\frac{1}{2}x + 4 & \\text{if } -2 < x \\leq 4 \\\\ -2x + 14 & \\text{if } x > 4 \\end{cases}$$

$

Explanation:

Step1: Find max of $f(x)$ (first piece)

For $x \leq -6$, $f(x)=\frac{5}{2}x+18$. This is an increasing function, so its maximum on this interval is at $x=-6$:
$\frac{5}{2}(-6)+18 = -15 + 18 = 3$

Step2: Find max of $f(x)$ (second piece)

For $-6 < x \leq -2$, $f(x)=3$, so the value is always 3.

Step3: Find max of $f(x)$ (third piece)

For $-2 < x \leq 4$, $f(x)=\frac{1}{2}x+4$. This is an increasing function, so its maximum on this interval is at $x=4$:
$\frac{1}{2}(4)+4 = 2 + 4 = 6$

Step4: Find max of $f(x)$ (fourth piece)

For $x > 4$, $f(x)=-2x+14$. This is a decreasing function, so its value is less than $-2(4)+14=6$.

Step5: Identify max of $f(x)$

Compare all values: $3, 3, 6, <6$. The maximum of $f(x)$ is 6.

Step6: Find max of $g(x)$ from graph

Looking at the graph of $g(x)$, the highest point has a $y$-value of 6.

Answer:

The maximum value of $f(x)$ is equal to the maximum value of $g(x)$ (both are 6).