Find the value or values of c that satisfy the equation f(b)-f(a)/b-a = f’(c) in the conclusion of the Mean Value Theorem for the function and intervals.

Question

Find the value or values of c that satisfy the equation f(b)-f(a)/b-a = f’(c) in the conclusion of the Mean Value Theorem for the function and intervals.

g(x) = {x³, -2 ≤ x ≤ 0; x², 0 < x ≤ 2

Answer 1

Hey Alex,

I'll help you to set up the first one, and then you can try the second on your own. In layman's terms, the Mean Value Theorem just says that when you have a continuous function on a closed interval, and you find the slope between the two end points, you're guaranteed some x value in between that will have a tangent line with that same slope.

So for our purposes, let's look at x³ on the interval -2 ≤ x ≤ 0.

We can start by finding the slope between our two end points. Our x values are -2, and 0, we get our y values to match those x's by plugging into the function. So our y's would be (-2)³ = -8, and (0)³=0. So the two end points are (-2,-8) and (0,0). Using the slope formula, then, we have the slope between our end points is 4.

Now we use that to find an x value in that window that gives us a derivative output of 4.

Take the derivative: 3x²

Set it equal to the slope we found: 3x² = 4.

Solve for x: x = ±2/√3.

Since we're only looking for the value between -2 and 0, we'll choose the negative. If you want, you can then rationalize by multiplying top and bottom by rad 3 to get:

x = -(2√3)/3.

I'll leave the next one to you. Feel free to reach out if you have any clarifying questions.