Consider the function f(x)=-2x^3+33x^2-108x+2. For this function, there are three important intervals: (-Inf,A], [A,B], [B,Inf) where A and B are the critical points. Find A and B and for each of the important intervals, tell whether f(x) is increasing or decreasing.
Get the derivative with respect to x:
f(x) = -2x^3 + 33x^2 - 108x + 2
f'(x) = -6x^2 + 66x - 108
Let f'(x) = 0 and get the values of x:
0 = -6x^2 + 66x - 108
0 = x^2 - 11x + 18
0 = (x - 2)(x - 9)
x = 2 and x = 9
Therefore, A = 2 and B = 9.
At (-Inf , 2] : decreasing (because f'(x) < 0 at these values of x)
At [2,9] : increasing (because f'(x) > 0 at these values of x)
At [9,+Inf) : decreasing (because f'(x) < 0 at these values of x)
Hope this helps :3
To find the critical points for the function f(x) = -2x^3 + 33x^2 - 108x + 2, we need to find the values of x where the derivative of the function is equal to zero.
Step 1: Find the derivative f'(x) of the function.
f'(x) = -6x^2 + 66x - 108
Step 2: Set f'(x) = 0 and solve for x.
-6x^2 + 66x - 108 = 0
Now, we can solve this quadratic equation. We can either factor it or use the quadratic formula.
Using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = -6, b = 66, and c = -108.
x = (-66 ± √(66^2 - 4(-6)(-108))) / (2(-6))
x = (-66 ± √(4356 - 2592)) / (-12)
x = (-66 ± √1764) / (-12)
x = (-66 ± 42) / (-12)
Simplifying further:
x1 = (-66 + 42) / (-12) = -24 / (-12) = 2
x2 = (-66 - 42) / (-12) = -108 / (-12) = 9
Therefore, the critical points of the function are x = 2 and x = 9, which represent the values of A and B, respectively.
Now, we can analyze each interval to determine whether f(x) is increasing or decreasing.
Interval (-∞, A]:
To determine the behavior of the function on this interval, we need to evaluate f'(x) for a value less than 2. Let's use x = 0.
f'(0) = -6(0)^2 + 66(0) - 108 = -108
Since the derivative is negative, f(x) is decreasing on the interval (-∞, A].
Interval [A, B]:
To determine the behavior of the function on this interval, we need to evaluate f'(x) for a value between 2 and 9. Let's use x = 5.
f'(5) = -6(5)^2 + 66(5) - 108 = 42
Since the derivative is positive, f(x) is increasing on the interval [A, B].
Interval [B, ∞):
To determine the behavior of the function on this interval, we need to evaluate f'(x) for a value greater than 9. Let's use x = 10.
f'(10) = -6(10)^2 + 66(10) - 108 = -408
Since the derivative is negative, f(x) is decreasing on the interval [B, ∞).
In summary:
- f(x) is decreasing on the interval (-∞, A] (x < 2)
- f(x) is increasing on the interval [A, B] (2 ≤ x ≤ 9)
- f(x) is decreasing on the interval [B, ∞) (x > 9)
To find the critical points and determine the intervals of increasing and decreasing for the given function f(x) = -2x^3 + 33x^2 - 108x + 2, we need to follow these steps:
Step 1: Find the derivative of f(x) using the power rule. Let's denote the derivative of f(x) as f'(x).
If f(x) = -2x^3 + 33x^2 - 108x + 2, then
f'(x) = d/dx (-2x^3) + d/dx (33x^2) - d/dx (108x) + d/dx (2).
Simplifying this, we get:
f'(x) = -6x^2 + 66x - 108.
Step 2: Set f'(x) equal to zero and solve for x to find the critical points.
To find the critical points, we solve the equation -6x^2 + 66x - 108 = 0.
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula.
Using the quadratic formula, x = [-b ± √(b^2 - 4ac)] / (2a), where a = -6, b = 66, and c = -108.
Substituting these values, we have:
x = [-66 ± √(66^2 - 4(-6)(-108))] / (2(-6)).
Simplifying further, we have:
x = [-66 ± √(4356 - 2592)] / (-12).
x = [-66 ± √(1764)] / (-12).
x = [-66 ± 42] / (-12).
Simplifying even more, we get two possible values for x:
x₁ = (-66 + 42) / (-12) = -24 / (-12) = 2.
x₂ = (-66 - 42) / (-12) = -108 / (-12) = 9.
Therefore, the critical points are A = 2 and B = 9.
Step 3: Analyze the intervals and determine whether f(x) is increasing or decreasing.
Now that we have the critical points, we can determine the intervals where the function is increasing or decreasing. To do this, we evaluate the sign of the derivative f'(x) in each interval.
Interval (-∞, A]:
Choose any number less than 2 and substitute it into f'(x). For example, let's pick x = 0.
f'(0) = -6(0^2) + 66(0) - 108 = -108.
Since f'(0) = -108 < 0, the derivative is negative, indicating that f(x) is decreasing in the interval (-∞, A].
Interval [A, B]:
Choose any number between 2 and 9 and substitute it into f'(x). For example, let's pick x = 5.
f'(5) = -6(5^2) + 66(5) - 108 = 312.
Since f'(5) = 312 > 0, the derivative is positive, indicating that f(x) is increasing in the interval [A, B].
Interval [B, ∞):
Choose any number greater than 9 and substitute it into f'(x). For example, let's pick x = 10.
f'(10) = -6(10^2) + 66(10) - 108 = -246.
Since f'(10) = -246 < 0, the derivative is negative, indicating that f(x) is decreasing in the interval [B, ∞).
In summary:
- The interval (-∞, A] (where A = 2) corresponds to f(x) being decreasing.
- The interval [A, B] (where A = 2 and B = 9) corresponds to f(x) being increasing.
- The interval [B, ∞) (where B = 9) corresponds to f(x) being decreasing.