Theory and Examples


[Technology Exercise] Graph the functions in Exercises 63–66. Then find the extreme values of the function on the interval and say where they occur.


h(x) = |x + 2| − |x − 3|, −∞ < x < ∞

Finding Extrema from Graphs

In Exercises 7–10, find the absolute extreme values and where they occur.

Finding Extrema from Graphs

In Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.

f(x) = |x|, −1 < x < 2

Finding Extrema from Graphs

In Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.

g(x) = {−x, 0 ≤ x < 1

x − 1, 1 ≤ x ≤ 2

Theory and Examples

[Technology Exercise] Graph the functions in Exercises 63–66. Then find the extreme values of the function on the interval and say where they occur.

f(x) = |x − 2| + |x + 3|, −5 ≤ x ≤ 5

For the function graphed below, which interval contains a local maximum? $[-1,2]$, $[2,4]$, $[4,6]$, or $[6,8]$?

For the function $f\left(x\right)$ defined on the interval $[0,6]$, the graph shows a peak at $x=2$ and a trough at $x=5$. Which interval contains the local maximum of $f\left(x\right)$?

Over which interval is the graph of $f\left(x\right)=\frac{1}{2}{x}^2+5x+6$ increasing?

For the function $f(x,y)=x^2+y^2-4x-6y+13$, which ordered pair is closest to a local minimum of the function?