f(x) -2x^3+2x^2-3x+2

Find the average slope of this function on the interval (–3–1) ________ <--A
By the Mean Value Theorem, we know there exists a c in the open interval (–3–1) such that f'(c) is equal to this mean slope. Find the value of c in the interval which works _________ <--B

A=?
B=?

ty

f(-3) = 54 + 18 + 9 + 2 = 83

f(-1) = -2 + 2 + 3 + 2 = 5

slope = (5-83)/(-1+3) = -39

f ' (x) = -6x^2 + 4x - 3

then -6x^2 + 4x - 3 = -39
6x^2 - 4x -36 = 0
3x^2 - 2x - 18 = 0
x = (2 ± √220)/6
= appr 2.8054 or appr -2.1388

To find the average slope of the function f(x) on the interval (-3, -1), we need to calculate the difference in the function values at the endpoints of the interval and divide it by the difference in the x-values of the endpoints.

A = (f(-1) - f(-3))/(-1 - (-3))

To find the value of f(x) at a specific x, substitute the value of x into the function equation.

For f(-1), substitute -1 into the function equation:
f(-1) = -2(-1)^3 + 2(-1)^2 - 3(-1) + 2

f(-1) = -2 + 2 + 3 + 2
f(-1) = 5

For f(-3), substitute -3 into the function equation:
f(-3) = -2(-3)^3 + 2(-3)^2 - 3(-3) + 2

f(-3) = -2(-27) + 2(9) + 9 + 2
f(-3) = 54 + 18 + 9 + 2
f(-3) = 83

A = (f(-1) - f(-3))/(-1 - (-3))
A = (5 - 83)/(-1 + 3)
A = (5 - 83)/2
A = -78/2
A = -39

So, the average slope of the function on the interval (-3, -1) is -39.

According to the Mean Value Theorem, there exists a point c in the open interval (-3, -1) such that f'(c) is equal to the mean slope, which is -39.

To find the value of c, we need to calculate the derivative of the function f(x) and solve for c.

f(x) = -2x^3 + 2x^2 - 3x + 2

To find f'(x), take the derivative of each term separately:
f'(x) = -6x^2 + 4x - 3

Now, we need to solve f'(c) = -39 for c:
-6c^2 + 4c - 3 = -39

Simplify the equation:
-6c^2 + 4c + 36 = 0

Now, we can solve this quadratic equation by factoring or using the quadratic formula. Since it doesn't factor easily, let's use the quadratic formula:

c = (-4 ± √(4^2 - 4(-6)(36))) / (2(-6))

c = (-4 ± √(16 + 864)) / (-12)

c = (-4 ± √880) / (-12)

c ≈ (2.37) or c ≈ (-2.03)

So, the value of c that works in the interval (-3, -1) is approximately 2.37 or approximately -2.03.

Therefore, A = -39 and B is approximately 2.37 or approximately -2.03.