Explain Rolle’s Theorem with a sketch.

Question

Accepted Answer

Below there. Today we're 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. Consider the function F of X is equal to x2 minus 4 x + 3. On the interval, square brackets 1.3, verify that Rawl's theorem applies to it, and if so, find the value of C that satisfies F C equals 0. Awesome. So it appears for this particular problem we're asked to solve for two separate. Answers. Our first answers are trying to verify that Raw's theorem can be applied to this particular function F of X, and if it can, we need to figure out the value is C that satisfies F equals 0. Awesome. So now that we know what we're solving for, let's read off our multiple choice answers to see what our final answer might be. So A is C equals 1.5, B is C equals 2. And C is C equals 2.5 and D is Ra's theorem does not apply. Awesome. So first off, in order to solve for this problem, we need to we need to recall the conditions that are required for Roll's theorem. So we need to recall a note that Rawl's theorem states that if a function F of X satisfies the following conditions on the interval, square brackets, A B, where let's take a moment here to pause and note just in case we can't recall off the top of our head. Square brackets indicate that this is a closed interval. If they were parentheses, that would mean it's an open interval, but this is a closed interval. So once again, Rawl's theorem states that if a function F of X satisfies the following conditions on the interval square brackets A, B, that means that it's going to be continuous. So let's make a note of this in red. So it's going to be continuous. So for a square brackets A, B, it's going to be continuous. I'll put a capital C for continuous. And then it's also important to note that parentheses A, B, it's going to be differentiable, which I just put a capital D for to denote different differentiable. And we also need to note that F of A. It's going to be equal to F of B. Fantastic. So, once again, these are the conditions that need to be met in order for Raw's theorem to be met or to be accepted. So, it's important to note that if all these conditions are met, That will mean, without a doubt that there exists some That C is an element of, and let's write this down, since this is an important note. So if these conditions are met for raw's theorem, that means that there exists some C, which is an element. Of square brackets A B such that F of C equals 0, where this prime symbol means that's the first derivative of this function of C, so F of C. Fantastic. So we need to note that we're told that we're given that F of X is equal to X2 minus 4 x + 3. On this closed interval of square brackets 1.3, fantastic. So we need to note and recall and recognize that F of X is a polynomial, which is continuous on this closed interval of 1.3. And also due to the fact that it's, it is a polynomial, that will mean that F of X is differentiable on this interval, which is open on the open interval. And we'll write this down at parentheses 1.3. So it is differentiable. Fantastic. Also, we need to note that F of 1 is equal to, and once again this one inside of the parentheses mean we're plugging in a value of 1 in for X. So when we do that, we'll take 1 squared. Minus 4 multiplied by 1 + 3, which will be equal to 0, and we also need to note that F of 3 is going to be equal to 3 squad minus 4 multiplied by 3 + 3, which will be equal to 0, and therefore F of 1 will be equal to F of 3 in this particular case. So therefore, without a doubt, Raw's theorem does apply to this particular function. So now, because we found out that Rah's theorem applies, we now need to find this value of C. So in order to find C, we need to find the first derivative of this function of F of X. So we need to differentiate F of X with respect to X, and we will find that F of X will be equal to 2 X minus 4. And now we need to set F C equal to 0, so we need to set F. Of C equal to 0. So what we need to do is we need to take Or we need to write that 2 minus. Or is equal to 0 based off of our expression that we found for F of X, but instead of X, we're going to replace X with a C, so we'll get 2 multiplied by C minus 4 is equal to 0. Now we need to rearrange this equation to isolate and solve for C all by itself, and when we do that, we will find that C is equal to 2, and that is our final answer. Hooray, we did it. So looking at our multiple choice answers, the correct answer without a doubt has to be multiple choice answer letter B, which once again is C equals 2. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Explain Rolle’s Theorem with a sketch.

Key Concepts

Rolle's Theorem

Continuity and Differentiability

Critical Points

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