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??
I found the derivative which is -6x^2+90x-300
-6(x^2-15x +50)
(x-5)(x-10) so I thought the critical points would be 5 and 10 but that's not what my book says??
Nevermind I got it!
To find the critical points of a function, you need to set the derivative of the function equal to zero and solve for x. In this case, you correctly found the derivative of f(x) to be -6x^2 + 90x - 300.
Next, you factored the derivative as -6(x^2 - 15x + 50). However, the factored form should be (x - 10)(x - 5) instead of (x - 5)(x - 10). This is a common algebraic mistake but a significant one in this case.
Setting the factored derivative equal to zero, you get:
(x - 10)(x - 5) = 0
Now, you can solve for x by setting each factor equal to zero:
x - 10 = 0 ---> x = 10
x - 5 = 0 ---> x = 5
So, the value of x at the critical points is 5 and 10. Therefore, A = 5 and B = 10.
Now, to determine whether each interval (-∞, A], [A, B], [B, ∞) is increasing or decreasing, you can plug in a test value from each interval into the derivative.
For the interval (-∞, A], you can choose a value less than 5, for example, x = 0. Plugging this value into the derivative -6x^2 + 90x - 300, you get:
-6(0)^2 + 90(0) - 300 = -300
Since the resulting value is negative, the function is decreasing in this interval.
For the interval [A, B], you can choose a value between 5 and 10, for example, x = 7. Plugging this value into the derivative, you get:
-6(7)^2 + 90(7) - 300 = 90
Since the resulting value is positive, the function is increasing in this interval.
For the interval [B, ∞), you can choose a value greater than 10, for example, x = 15. Plugging this value into the derivative, you get:
-6(15)^2 + 90(15) - 300 = -300
Once again, the resulting value is negative, indicating that the function is decreasing in this interval as well.
To summarize:
A = 5
B = 10
(-∞, A] is decreasing
[A, B] is increasing
[B, ∞) is decreasing
I hope this explanation clarifies the concepts and helps you understand the correct critical points and the intervals of increasing and decreasing behavior of the function.