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.

b. f(x) = x, g(x) = x², (a, b) arbitrary

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Let F of X equal X 2 and G of X equal X 3. On the interval, square brackets of A B with a less than B use Cauchy's mean value theorem to find all values of C is an element of parenthesis of AB that satisfy the conclusion of the theorem. OK. So it appears for this particular problem we're given expressions for the function F of X and G of X, and we're also given a specific interval, which is a closed interval of A B where A is less than B, and we're asked to use all this information and we're asked to also recall and use Cauchy's mean value theorem to find all values of C where C is an element. Of the open interval means two parentheses means that it's an open interval for A, B, that satisfy the conclusion of the theorem. So now that we know what we're ultimately trying to solve for, we must first recall and state Cauchy's mean value theorem formula, which states that parentheses F of B minus F of A. Multiplied by G C will be equal to drumroll G B minus it's gonna be G of B minus G of A. Multiplied by F of C. So F of C. So now what we need to do is we need to compute the differences. So we need to take F of B minus F of A is equal to B2 minus A2, which will be equal to parentheses B minus A multiplied by B plus A. By the difference of squares, which will mean that g of B. Minus G of A will be equal to B 3 minus 3, which will be equal to parentheses B minus A multiplied by parenthesis of B2 plus a multiplied by B plus A to the power of 2. By factoring a cubic difference. OK. So, with that said, we now need to begin computing the derivatives. So F of X will be equal to 2 X, which will mean that F of C will be equal to 2C. And G of X will be equal to 3 x 2, which will mean that G of C will be equal to 3, multiplied by C 2 or 3C to 2. OK. So now we need to substitute our values back into our formula. So we need to take parentheses B minus A, so we need to take parentheses B minus A. Multiplied by, so once again parentheses B minus A multiplied by B plus A multiplied by 3 by 3 C to the power 2, which is equal to parentheses B minus A multiplied by parentheses B 2 plus A multiplied by B plus A 2 multiplied by 2 C. And we now need to cancel the common factor parentheses B minus A, because, or since A cannot equal B, which will give us 3 multiplied by C2 multiplied by parenthesis of B plus A, which will be equal to 2 multiplied by C or 2C multiplied by parentheses B 2 plus A multiplied by B. Plus A to the power of 2. So now, what we need to do is we need to rearrange our expression to factor out the value of C. So we need to take 3 or 3 multiplied by C to the power of 2, multiplied by parentheses B plus A. M 2 C. Multiplied by parenthesis of B to the power of 2. Plus, A multiplied by B plus A to the power of 2, which will be simply equal to 0. Which when we factor out our C, we'll find that C multiplied by the parentheses of 3C multiplied by the parenthesis of B plus A minus 2 multiplied by the parenthesis of B 2 plus A multiplied by B plus A to the power of 2 will be equal to 0. So now we need to solve our factor equation. So we will find that C is equal to 0 or 3 multiplied by C, multiplied by parenthesis of B plus A. Minus 2 multiplied by parenthesis of be to the power of 2. Plus, A multiplied by B plus A to the power of 2. So once again, We need to note that it will be this expression will be equal to 0 due to our prior knowledge of the zero product property. And we'll also note that 3 or 3 multiplied by C multiplied by parenthesis of B plus A will be equal to 2 multiplied by B 2 + A multiplied by B plus A2 will mean that C is also equal to 2 multiplied by parenthesis of B 2 plus A multiplied by B plus a 2. All divided by 3 multiplied by parenthesis of B plus A. Fantastic. So that means our two final answers without a doubt will be C is equal to 0 due to the zero product property, and C will also be equal to 2 multiplied by the parenthesis of B 2 plus A multiplied by B plus A 2 all divided by 3 multiplied by the parenthesis of B plus A. And that's it. We've officially solved for this problem. So to quickly recap what we've learned, our final two answers without a doubt will be as follows. So once again, our first answer for C will be 0, and our Second answer for C will be that C is equal to 2 multiplied by parenthesis of B 2 plus a multiplied by B plus a 2, all divided by 3 multiplied by the parenthesis of B plus A. And that's it. That is our final answer pair. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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

Watch next

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