Question

Given f '(x) = (5 - x)(8 - x), determine the intervals on which f(x) is increasing or decreasing.

Answer 1

increasing decreasing?
So we need to determine where the function "turns" from increasing to decreasing and all that jazz, ya?

We call those locations "critical" or "stationary" points.

Answer 2

critical points exist where the first derivative is zero.
\[\large\rm 0=(5-x)(8-x)\]

Answer 3

So what values do you get for critical points? :)
We can construct a first derivative test once you've determined those points.

Answer 4

5 and 8?

Answer 5

\(\color{#000000 }{ \displaystyle 0>(5-x)(8-x) }\)
when f decreases (negative slope)

\(\color{#000000 }{ \displaystyle 0<(5-x)(8-x) }\)
when f increases (positive slope)

Answer 6

solve each thing for x.