Explain the Mean Value Theorem with a sketch.

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Suppose that F of X equals X2 + 3 X is the mean value theorem applicable on the interval from -2 to 1, explained by providing a graph. I've actually provided the graph on the right side here. I'll explain it as we go along. But first, we need to look at what the mean value theorem tells us. Mean value theorem. First tells us That a function To be satisfied With continuity On a close interval. From A to B. The function Is differentiable On the open interval From A to B. And if these conditions are true, There will then exist. Some see value In the interval, From A to B Such that F privacy. This equals to F of B minus F of A divided by B minus A. So now let's check the conditions with our graph. Well, we can first see that this is a polynomial function. Polynomials are always continuous. And in this case, this is differentiable. And polynomials are actually always differentiable. So this satisfies both of our conditions. So now, we can check our F of C to see if the mean value theorem is satisfied. So we have F C. Equals F of B. Minus F of A Divided by B minus A. Where we have F. 1 minus F of -2. Divided by 1 minus -2. We can plug in one to our equation. That gives us 4. We plug in -2. We get -2. All divided by 3. We get 6 divided by 3. Which equals 2. We now have FC. Equals 2. Now we'll take our derivative. 2C + 3. Set equals 2. And then solve for seat 2 C equals -1, and C equals -12. This is in the interval. From 2 to 1. Which confirms the mean value theorem. If we were to look at our graph, We see that C equals neither 1/2 is on this tangent line. And we can also see That our secret line between the two points. Is the same as our tangent line. We can then confirm that the mean value theorem is true. This means the answer to our problem is answer C. The mean value theorem is applicable because FX is continuous and differentiable on the interval -2 to 1. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Explain the Mean Value Theorem with a sketch.

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