Consider the function

f(x)=–3x3–1x2+1x+1Find the average slope of this function on the interval (–2–1).
By the Mean Value Theorem, we know there exists a c in the open interval (–2–1) such that f(c) is equal to this mean slope. Find the value of c in the interval which works

I found the mean which is -17 but I can't get the C. I don't know what im doing wrong

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

f(-2) = 19
slope = (2-19)/1 = -17 agree

f'(x) = -9 x^2 - 2 x + 1
for what values of x is that -17?
-9 x^2 - 2 x + 1 = -17

9 x^2 + 2 x - 18 = 0

x = [-2 +/- sqrt (4+648)]/18
= 1.3 or -1.53

-1,53 is in the domain

check it by calculating
f'(1.53)

To find the average slope of a function on a given interval, you need to find the difference in function values between the endpoints of the interval, and divide it by the difference in the x-values of the endpoints. In this case, you are given the interval (-2, -1).

To begin, let's find the difference in function values between the endpoints of the interval. We need to evaluate the function at -2 and at -1.

f(-2) = -3(-2)^3 - 1(-2)^2 + 1(-2) + 1
= -24 - 4 - 2 + 1
= -29

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

The difference in function values is: f(-1) - f(-2) = -2 - (-29) = 27.

Now, let's find the difference in the x-values of the endpoints: -1 - (-2) = 1.

The average slope is calculated by dividing the difference in function values by the difference in x-values: average slope = (f(-1) - f(-2))/(1) = 27/1 = 27.

According to the Mean Value Theorem, there exists a value c in the open interval (-2, -1) such that f'(c) is equal to the average slope. To find the value of c, we need to find the derivative of the function f(x).

f'(x) = -9x^2 - 2x + 1

To solve for c, we can set f'(c) equal to the average slope and solve for c:

-9c^2 - 2c + 1 = 27

This is a quadratic equation, so we can solve it by setting it equal to zero:

-9c^2 - 2c + 1 - 27 = 0

-9c^2 - 2c - 26 = 0

Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. In this case, the quadratic equation is not easily factorable, so we will use the quadratic formula:

c = (-(-2) ± √((-2)^2 - 4(-9)(-26)))/(2(-9))

Simplifying this expression gives:

c = (2 ± √(4 - 936))/(-18)

c = (2 ± √(-932))/(-18)

Since the discriminant (the term under the square root) is negative (√(-932) is not a real number), there are no real solutions for c in the interval (-2, -1) that satisfy the equation f'(c) = 27.

Therefore, there is no value of c in the interval (-2, -1) that works for this problem.