80. Find all values of c that satisfy the conclusion of Cauchy...

Question

80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.

a. f(x) = x, g(x) = x², (a, b) = (-2, 0)

Accepted Answer

Welcome back everyone. Let f x be equal to x 2 and g x be equal to x cubed on the interval from 1 to 3 noninclusive. Find all values of c guaranteed by Cauchy's mean value theorem. A says 5 divided by 3, B 7 divided by 3, C 11 divided by 6, and D 13 divided by 6. So for this problem, let's remember Quoci's mean value theorem. It states that there exists a constanty in the interval. From 1 to 3. Non-inclusive with F of 3 minus F of 1. Multiplied by g at c and this must be equal to g of 3 minus g of 1. Multiplied by The derivative. F At See So what we want to do is simply identify each parameter. Let's begin with F of 3. F of 3 is the value of the function F1 x is 3, so that's 3 squared, which is 9. Followed by f of 1, which is 1 squared, and that's 1. Now we want to identify G in a general form. So G. Of x is the derivative of g of x, and that's the derivative of x cubed according to the power rule that's 3 x squared. So now g at c is simply equal to 3 c square. Then we want to find g of 3, which is the value of g of x when x is 3 and that's 3 cubed. Which is 27. We also want to find G of 1, which is 1 cubed, and that's 1. And finally, we want to find F C. So let's begin by finding F. Of x that's the derivative of f of x. The derivative of x squared is 2 x. So f at c is simply 2 c. Now let's go ahead and substitute all of these into our equation, which gives us. F of 3 minus f of 1, that's 9 minus 1, which is 8. Followed by g at c, which is 3 c square, and this must be equal to g of 3 minus g of 1. So that's 27 minus 1, which is 26. Multiplied by F at C, which is 2C. So we are going to get 24 c squared equals 52C. From here, what we can do is simply combine our terms on the same side. So that'd be 24C 2 minus 52C equals 0. Now, we can divide both sides by 4 so that The equation can be simplified into 6. C 2 minus 13 c equals 0 and we can factor out c, so that's c multiplied by 6 cminus 13 equals 0. And to solve this equation, we essentially have to notice that there are two possible solutions. Either c is equal to 0 according to the zero product property, right, or 6c minus 13 is equal to 0. So what we can do is simply exclude c equals 0 because c is between 1 and 3, and we can focus on the final equation. So 6 c must be equal to 13 and therefore c is equal to 13 divided by 6, which is a number between 2 and 3, right? So it is within the interval and it is our solution. So the answer to this problem corresponds to the answer to d. Thank you for watching.

80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.a. f(x) = x, g(x) = x², (a, b) = (-2, 0)

Watch next

80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.
a. f(x) = x, g(x) = x², (a, b) = (-2, 0)