Find the absolute extreme values of the function on the interval.
-$$f ( x ) = | x - 7 | , \quad 5 \leq x \leq 10$$
A) absolute maximum is 3 at $x = 10$; absolute minimum is 2 at $x = 5$
B) absolute maximum is 2 at $x = 5$; absolute minimum is 0 at $x = 7$
C) absolute maximum is $- 2$ at $x = 5$; absolute minimum is 3 at $x = 10$
D) absolute maximum is 3 at $x = 10$; absolute minimum is 0 at $x = 7$

Question

Quizplus · Accepted Answer

To find the absolute extreme values of the function, we need to check the function values at the endpoints and at the critical points (where the derivative is undefined or equal to zero).

Here, the endpoints are $x=5$ and $x=10$.

At $x=5$, $f(x) = |5-7| = 2$.

At $x=10$, $f(x) = |10-7| = 3$.

So, we have found the absolute minimum and maximum values of $f$ at the endpoints.

Now we need to check if there are any critical points in the interval $(5,10)$.

The function $f(x)=|x-7|$ has a critical point at $x=7$. However, $x=7$ is not in the interval $(5,10)$. Therefore, $x=7$ is not a critical point in this case.

Since the absolute minimum and maximum values of $f$ occur at the endpoints, we can conclude that:

The absolute minimum value is 2 at $x=5$, and

the absolute maximum value is 3 at $x=10$.

Therefore, the answer is D: absolute maximum is 3 at $x=10$ and absolute minimum is 2 at $x=5$.

Find the Absolute Extreme Values of the Function on the Interval