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.

g(x) = x³ - x² - 5x - 3; [-1, 3]

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 X cubed minus 3X2 minus 6 X + 8 on the closed interval from 1 to 4. 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're given 4 possible choices as are. For choice A, we have X is equal to 1 minus the square root 3, and X is equal to 1 plus the square root of 3. For choice B, we have X is equal to minus 1 plus the square root of 3. For choice C, we have X as equal to 1 plus the square root of 3. And for choice D, we have the rules theorem does not apply to G of X only give intervolt. Now, the first part of this problem asks us to determine if Roe's theorem is applicable to our function on the given interval. To recall that for rules theorem to be applicable, we need to satisfy three conditions. The first condition is that G of X needs to be continuous. On the close interval from 1 to 4. So if we look at our function G of X, we notice that we have a polynomial. Now polynomials are continuous over the entire real line, so therefore G of X is continuous over the entire real line. And so therefore G of X is continuous on the close interval from 1 to 4. That means that our first condition is met. Now, for the second condition, we need G of X to be differentiable over the open interval from 1 to 4. And so, again, since we have a polynomial for G of X, polynomials are differentable everywhere on the real line, and so therefore G of X is also differentiable over the entire real line. Therefore, GFX is differentiable on the open interval from 1 to 4, and so therefore the Second condition is met. Now, our third condition states that G of X has to evaluate to the same value at the two endpoints of our interval. So we'll need G 1 B equal to G of 4. So we'll just calculate those two quantities. So 4 G of 1. This is going to be equal to. When cued -3 multiplied by 1 squared. 6 Multiplied by 1. Plus 8. And when we evaluate this expression, this is going to be equal to 0. 4G of 4, this is going to be equal to. 4 cubed -3 multiplied by 4 squared. -6 multiplied by 4. Plus 8, and evaluating this expression, this is also going to be equal to 0. So since these two values are the same, that means that our third condition is met. So therefore, Roe's theorem is applicable to our function over the given interval. So now we need to use Rohl's theorem to find the points that are guaranteed by Rohl's theorem. So, we call that for Rohl's theorem, it says that there is a point X, that's that. G of X is equal to 0. On the open interval from 1 to 4. So the first step We need to do is to determine G of X. So G of X. It's going to be equal to the derivative with respect to X. Of the quantity of X cubed minus 3 X 2d. Minus 6 X Plus 8. So taking this derivative, this is going to be equal to 3 X2. Minus 6 X -6. So we'll need to set this expression equal to 0, so we'll have 3 x 2 minus 6 X minus 6 is equal to 0. So now we're going to need to solve this for X. The first thing I notice is that I can divide both sides of my equation by 3. In doing so, I'll get X2 minus 2 X. minus 2 is equal to 0. And here we have a quadratic equation, and so we can find the values for x using our quadratic formula. So we call for the quadratic formula that we're going to have X. It's equal to -2. Plus or minus the square roots of the quantity. Of -2. Weird? -4, multiplied by 1, multiplied by -2 in quantity. And that whole quantity is divided by the quantity of 2 multiplied by 1. So simplify this expression, this is going to be equal to 2, plus or minus. The square root of the quantity of 4. Plus 8 Inwati That whole quantity is divided by 2. So simplifying this expression, this is going to be equal to. 1, plus or minus the square root of 3. So this gives us two solutions. We have X is equal to 1 plus the square root of 3. Which is approximately equal to. 2.73. And we have X is equal to 1 minus the square root of 3, which is approximately equal to -0.73. So we need X to be within the open interval from 1 to 4, and only one of the X values satisfies that condition, which is our first X value, which is X is equal to 1 plus the square root of 3. So that is the point that is guaranteed by R's theorem, and so that is going to be the answer to our question. And so if we come back up to our answer choices, we see that the correct answer choice corresponds to answer choice D. 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.

g(x) = x³ - x² - 5x - 3; [-1, 3]

Key Concepts

Rolle's Theorem

Continuity

Differentiability

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