Question

Given f '(x) = (2 - x)(6 - x), determine the intervals on which f(x) is increasing or decreasing.

Answer 1

The derivative can determine the increase or decreasing motion of a function, in order to know these regions you derivate the function and then study the sign of the derivative.

When the derivative is positive, then the function is increasing
when it's negative, it's decreasing.
So, having the function \[g'(x)=(2-x)(x-6)\]
You can observe it is already the derivative of a function, so we just want to study the sign of it.

Answer 2

ok these are my answer choices Decreasing (-∞, 2); increasing on (6, ∞)
Decreasing (2, 6); increasing on (-∞, 2) U (6, ∞)
Decreasing (-∞, 2) U (6, ∞); increasing on (2, 6)
Increasing (-∞, -2) U (-6, ∞); increasing on (-2, -6)
but from what you said wouldnt both have to be decreasing bc like you said its negative

Answer 3

Yes, the derivative value should be, do not confuse it with the xalue of "x", the value that has to be negative in order to conclude that a function is decreasing in a given interval is "f'(x)".

Answer 4

ok so is decreasing (-∞, 2) U (6, ∞) and increasing on (2, 6)

Answer 5

You can give it a try by plotting an x value higher than 6 and see what sign the result yields.
Same goes for one lower than 2.

Answer 6

You are correct

Answer 7

thank you !