11–18. Rolle’s Theorem Determine whether Rolle’s Theorem appli...

Question

11–18. Rolle’s Theorem Determine whether Rolle’s Theorem applies to the following functions on the given interval. If so, find the point(s) guaranteed to exist by Rolle’s Theorem.

ƒ(x) = sin 2x; [0, π/2]

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says consider the function G of X is equal to cosine of 3 X on the closed interval from minus pi divided by 6 to pi divided by 6. Determine if Rohl's theorem applies to this function on the given interval. If it does, find the points guaranteed by Rohl's theorem. We give 4 possible choices as our answers. For choice A, we have X's equal to 0. for choice B, we have X's equal to minus pi divided by 12, and X is equal to 0, and X is equal to pi divided by 12. For choice C, we have X is equal to minus pi divided by 12, and X is equal to pi divided by 12. And for choice D, the Rose theorem does not apply to G of X on the given interval. Now, the first part of this question, wants to determined if R's theorem is applicable to our function on the given interval. So, for rules theorem to be applicable, we need to satisfy three conditions. The first condition is that our function G of X has to be continuous on the close interval from minus pi divided by 6 to pi divided by 6. And if we look at our function G of X, we see that it is equal to cosine of 3 X. Recall that our cosine function is continuous over the entire real line. So that means that GF X here is also continuous over the entire real line. Therefore, G of X is continuous only closed interval from minus pi divided by 6 to pi divided by 6, and so our first condition is met. Now, our second condition is that G of X must be differentiable. Over the open interval from minus pi divided by 6 to pi divided by 6. So, in this case, we again look at our function G of X, and we notice that it is again equal to cosine of 3 X. Our cosine function is differentiable over the entire real line, so that means that G of X is differentiable over the entire real line. So G of X is differentiable on the open interval from minus pi divided by 6 to pi divided by 6. So that means that our second condition is also met. Now our third condition Is that our function G of X must evaluate to the same value at the two endpoints of our interval. So that means that G minus pi divided by 6 has to be equal to G of pi divided by 6. So we will just calculate those two quantities. For G of minus. Pi divided by 6. This is going to be equal to cosine of the quantity of 3 multiplied by minus pi divided by 6 in quantity, and this is going to be equal to cosine of minus pi divided by 2, which is equal to 0. We also need to evaluate G of pi divided by 6, and this is going to be equal to cosine of the quantity of 3 multiplied by pi divided by 6 in quantity. This is going to be equal to cosine of pi divided by 2, which is equal to 0. So both of these quantities evaluate to 0. That means that our third condition is also met. So Roz's theorem is applicable to our function over this interval. So, We need to find the points that are guaranteed by Roll's theorem. So recall for Roll's theorem that there is a point. X such that the of X. is equal to 0 on the open interval from minus pi divided by 6 to pi divided by 6. So, the first thing we need to do is take a derivative of our function GFX. So, that means we'll have G of X is equal to the derivative with respect to X. Of cosine of 3 X. And when we take a derivative of this function, this is going to be equal to -3, multiplied by sine of 3 X. So here we call that the derivative of our cosine function is minus sine, and we also had to apply our chain rule. So now substituting that that back into our expression for um Ro's theorem, so G of X is equal to 0, so that means that -3, multiplied by sine of 3 X. is equal to 0. Now, in order for the statement to be true, we need our sine function to be equal to 0, and we call that our sine function is equal to 0 whenever its argument is equal to an integer multiple of pi. That means that 3 X has to be equal to n multiplied by pi. So here n is an integer. And so we'll solve this for X by dividing both sides of the equation by 3, and we'll get X is equal to n multiplied by pi divided by 3. So now we need to find values for X that fall within our interval. So we'll start off by looking at N equal to 0. So when N is equal to 0, that means that X is also equal to 0, and that falls within our open interval from minus pi divided by 6 to pi divided by 6. If you look at the case of N equals to 1, That means that X is going to be equal to pi divided by 3. However, pi divided by 3 is greater than pi divided by 6, so this value falls outside of our interval. And any values of N greater than 1 are also going to fall outside the interval. So now if we look at the other direction with N equal to -1, we see that X is going to be equal to minus pi divided by 3. And minus pi divided by 3 is actually less than minus pi divided by 6, so it also falls outside the interval, and any um Value of N that is more negative than -1, is also going to have its X value fall outside of the interval. So that means that the only value that falls within our interval, and it's guaranteed by Rohl's theorem, is X equal to 0. So that is the answer to this question, and so if we look at our answer choices, we see that the correct answer choice corresponds to answer choice A. So I hope that this explanation has been useful, and I'll see you in the next video.

11–18. Rolle’s Theorem Determine whether Rolle’s Theorem applies to the following functions on the given interval. If so, find the point(s) guaranteed to exist by Rolle’s Theorem.ƒ(x) = sin 2x; [0, π/2]

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11–18. Rolle’s Theorem Determine whether Rolle’s Theorem applies to the following functions on the given interval. If so, find the point(s) guaranteed to exist by Rolle’s Theorem.

ƒ(x) = sin 2x; [0, π/2]