CPM Homework Help : CALC Problem 8-81

Multiple Choice: The maximum value of $f(x) = x(x − 3)(x − 7)$ on the interval $0 ≤ x ≤ 7$ is nearest to:

$12$

$14$

$16$

$22$

Hint 1:

Expand f before differentiating.

Hint 2:

Remember that a maximum value is a $y$ -value, not an $x$ -value.

Hint 3:

Notice that this is a CLOSED domain: $0 ≤ x ≤ 7$ .
Since maxima can exist where $f^\prime(x) = 0$ or where $f^\prime(x) =$ DNE, it is important to check the $y$ -values at each endpoint.
$f(0) =$ ________ and $f(7) =$ ________
After all, an endpoint might exceed the $y$ -values where $f^\prime(x) = 0$ ; in which case, that endpoint will be in the global maximum.