8.
f(x)= - 2x^3 + 45x^2 -300x +9
There are 3 intervals (-inf,A], [A,B], [B, inf) where A and B are critical points.
Find A and B?
For each of the intervals is it decreasing or increasing??
To find the critical points in the given function, we need to find the values of x for which the derivative of f(x) equals zero.
Step 1: Calculate the derivative of f(x) with respect to x:
f'(x) = -6x^2 + 90x - 300
Step 2: Set f'(x) equal to zero and solve for x:
-6x^2 + 90x - 300 = 0
Step 3: Factorize the quadratic equation:
-6(x^2 - 15x + 50) = 0
Step 4: Solve for x by factoring or using the quadratic formula. In this case, we can factorize the quadratic equation:
-6(x - 10)(x - 5) = 0
Setting each factor equal to zero:
x - 10 = 0 => x = 10
x - 5 = 0 => x = 5
So, A = 5 and B = 10 are the critical points.
To determine if the function is decreasing or increasing in each interval, we need to evaluate the derivative f'(x) at a point within each interval.
Interval 1: (-∞, A]
Choose a value less than 5, such as x = 0. Substitute this value into f'(x):
f'(0) = -6(0^2) + 90(0) - 300 = -300
Since f'(0) is negative, the function is decreasing in the interval (-∞, A].
Interval 2: [A, B]
Choose a value between 5 and 10, such as x = 8. Substitute this value into f'(x):
f'(8) = -6(8^2) + 90(8) - 300 = 72
Since f'(8) is positive, the function is increasing in the interval [A, B].
Interval 3: [B, ∞)
Choose a value greater than 10, such as x = 15. Substitute this value into f'(x):
f'(15) = -6(15^2) + 90(15) - 300 = -300
Since f'(15) is negative, the function is decreasing in the interval [B, ∞).
To summarize:
- In the interval (-∞, A], the function is decreasing.
- In the interval [A, B], the function is increasing.
- In the interval [B, ∞), the function is decreasing.