To determine the intervals on which the function f(x) is increasing or decreasing, we need to find the critical points. Critical points occur where the derivative f'(x) is either zero or undefined.
First, let's find the critical points by setting f'(x) equal to zero:
(2 - x)(6 - x) = 0
This equation will be satisfied when either (2 - x) = 0 or (6 - x) = 0.
1) When (2 - x) = 0, solving for x gives x = 2.
2) When (6 - x) = 0, solving for x gives x = 6.
So, we have two critical points: x = 2 and x = 6.
To determine the intervals of increase and decrease, we can test the sign of the derivative (f'(x)) on each interval. If the derivative is positive, f(x) is increasing. If the derivative is negative, f(x) is decreasing.
Now, let's test the sign of f'(x) on different intervals:
For x < 2, pick a value less than 2 (e.g., x = 0):
f'(0) = (2 - 0)(6 - 0) = 12 > 0
So, f'(x) is positive for x < 2, which means f(x) is increasing on the interval (-∞, 2).
For 2 < x < 6, pick a value between 2 and 6 (e.g., x = 4):
f'(4) = (2 - 4)(6 - 4) = -4 < 0
So, f'(x) is negative for 2 < x < 6, which means f(x) is decreasing on the interval (2, 6).
For x > 6, pick a value greater than 6 (e.g., x = 8):
f'(8) = (2 - 8)(6 - 8) = 12 > 0
So, f'(x) is positive for x > 6, which means f(x) is increasing on the interval (6, ∞).
Therefore, the correct option is:
1) Decreasing (-∞,2); increasing on (6,∞)