Suppose the derivative of a function f is f′(x)=(x−8)^7(x−1)^4(x+19)^8.
Then the function f is increasing on the interval what?
f(x) is increasing if f'(x) is positive
so (x−8)^7(x−1)^4(x+19)^8 > 0
we know that (x−1)^4(x+19)^8 is always positive since we have even exponents.
So all we have to do is look at
(x-8)^7 > 0
if x > 8 then (x-8)^7 > 0
if x < 8 it is negative
so the function is increasing for
x > 8
However, when x = 1 or -19, f'(x) = 0, so the function would not be increasing, but those two values would be excluded anyway by x > 8
so it is increasing for all real values of x such that
x > 8
To determine where the function f is increasing, we need to analyze the sign of the derivative f'(x).
Given that f'(x) = (x-8)^7 * (x-1)^4 * (x+19)^8
To determine the sign of the derivative, we need to consider the sign of each factor.
1. (x-8)^7:
- This factor will be positive for x > 8 and negative for x < 8.
2. (x-1)^4:
- This factor will be positive for x > 1 and negative for x < 1.
3. (x+19)^8:
- This factor will be positive for all values of x.
Now, let's consider the overall sign of the derivative:
- If all three factors have the same sign, then the overall sign will be positive.
- If two factors have one sign and the remaining factor has the opposite sign, then the overall sign will be negative.
- If one factor is zero, then the derivative will be zero at that specific value of x.
From the analysis above, we can conclude that the function f will be increasing in the intervals where all three factors have the same sign, which means:
1. For x > 8, both (x-8)^7 and (x-1)^4 are positive.
2. For x < 1, both (x-8)^7 and (x+19)^8 are negative.
Therefore, the function f is increasing on the interval x > 8.
To determine on which interval the function f is increasing, we need to analyze the sign of its derivative f'(x).
Given that f'(x) = (x-8)^7(x-1)^4(x+19)^8, we can determine the interval where f is increasing by examining the intervals where f'(x) is positive.
To do this, we need to consider the factors of f'(x) individually and analyze their signs:
1. (x-8)^7: This factor will be positive for all x > 8 and negative for all x < 8. Since this factor is raised to an odd power (7), its sign will not change when crossing zero (x=8).
2. (x-1)^4: This factor will be positive for all x > 1 and negative for all x < 1. Since this factor is raised to an even power (4), its sign will change when crossing zero (x=1).
3. (x+19)^8: This factor will be positive for all x > -19 and negatives for all x < -19. Since this factor is raised to an even power (8), its sign will not change when crossing zero (x=-19).
Now, let's analyze the combined signs of these three factors:
- When x < -19, all three factors are negatives. So, f'(x) is negative.
- Between -19 and 1, the first and third factors are positive, while the second factor is negative. So, f'(x) is negative.
- Between 1 and 8, the first and second factors are positive, while the third factor is negative. So, f'(x) is positive.
- When x > 8, all three factors are positive. So, f'(x) is positive.
Based on these analyses, we can conclude that the function f is increasing on the interval x ∈ (1, 8).