Find the open intervals on which f(x) = -6x^2 + 96x + 7 is increasing or decreasing.
a. increasing on (-inf, 16); decreasing on (16, inf).
b. increasing on (-inf, 14); decreasing on (14, inf).
c. increasing on (-inf, 84); decreasing on (84, inf).
d. increasing on (-inf, 56); decreasing on (56, inf).
e. increasing on (-inf, 8); decreasing on (8, inf).
I think the answer is e, but the answer is still greater than 0 when x = 10 is used?
this is a parabola
since we have a negative coef of x^2
it opens down (sheds water)
the real question is, where is the vertex?
complete the square
6 x^2 -96 x = 7
x^2 - 16 x = 7/16
x^2 -16 x + 64 = 64 7/16
(x-8)^2
vertex at x = 8
Yes, you are right , e.
what is f(8)?
-6x^2 + 96x + 7 =-6(64) +96(8) + 7
= -6(64) +12(64) + 7
= 6(64) + 7
= 391
now you say f(10) is bigger?
-6(100) +96(10) + 7
-600+960+7
367, nope , disagree
To determine the intervals on which the function is increasing or decreasing, we need to find the critical points and test the intervals between them. The critical points occur when the derivative of the function is equal to zero or undefined.
Given the function f(x) = -6x^2 + 96x + 7, let's find its derivative:
f'(x) = -12x + 96
To find the critical points, we set f'(x) = 0:
-12x + 96 = 0
Solving for x gives:
x = 8
Therefore, we have one critical point at x = 8.
Now, let's test the intervals to the left and right of the critical point. We can choose a value within each interval and evaluate the sign of f'(x) to determine whether the function is increasing or decreasing.
Interval (-∞, 8):
Choose x = 0 (within the interval)
f'(0) = -12(0) + 96 = 96
Since f'(0) > 0, the function is increasing on the interval (-∞, 8).
Interval (8, ∞):
Choose x = 10 (within the interval)
f'(10) = -12(10) + 96 = -24
Since f'(10) < 0, the function is decreasing on the interval (8, ∞).
Therefore, the correct answer is e. The function is increasing on the interval (-∞, 8) and decreasing on the interval (8, ∞).
To determine the intervals on which the function f(x) = -6x^2 + 96x + 7 is increasing or decreasing, we need to find the first derivative of the function and analyze its behavior.
First, find the derivative of f(x) by applying the power rule of differentiation. The derivative of -6x^2 + 96x + 7 is -12x + 96.
Now, set the derivative equal to 0 and solve for x.
-12x + 96 = 0
-12x = -96
x = 8
Next, choose test points on either side of the critical value x = 8 and evaluate the sign of the derivative at those points.
For x < 8, choose x = 0:
-12(0) + 96 = 96
Since the derivative is positive, f(x) is increasing on the interval (-∞, 8).
For x > 8, choose x = 10:
-12(10) + 96 = -24
Since the derivative is negative, f(x) is decreasing on the interval (8, ∞).
Therefore, the correct answer is e. The function is increasing on the interval (-∞, 8) and decreasing on the interval (8, ∞).