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Problem 1 Medium Difficulty

The graph of a function $ f $ is shown. Verify that $ f $ satisfies the hypotheses of Rolle's Theorem on the interval $ [0, 8] $. Then estimate the value(s) of $ c $ that satisfy the conclusion of Rolle's Theorem on that interval.

Answer

$x=2 \quad x=5$

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Video Transcript

Okay. Ah, the question here is asking. The graph of function F is shown verified that that verified that f satisfies the hypothesis of rolled storms on the interval doing through eight then estimate the values of sea that satisfied the conclusion of Rose Damned on that inner voice. So here I've drawing the function f and essentially there are three conditions for roll stare. So the first condition as explicit here is that if this continues on the clothes animal from zero to eight so what it means to be on close interval is that it includes the endpoint zero and eight which here it is obviously continues because you can essentially draw this function without picking up her pencil or that the limit at all these points exists. So that is continuous. We got that for difference. The second condition is after defensible on zero to eight on the open animal. Whether that's so does not include the the endpoint. Because if it did include the endpoint, technically the derivative don't exist because to function stop here and essentially like a curve. I mean, sharp point, so we don't include the endpoint. But since his functions smooth, we can also assume different ability. So it is defensible arm zero to eight and the last condition is that the end point of dysfunction? Um equal each other. So essentially here at a zoo accessory hurry after three and at eight, it is all for three. So we know that this is this is a true statement. No, these are just the conditions to satisfy the role still. So what is the row still actually tell Once used to recondition drastically satisfied. It also tells us that there is a sea in after Becks such that in the interval A to B which in this case, it zero to eight such that f prime of sissy. So one of these numbers see equally dear. So what does that mean? Well, that means that the tangent line I wanted these in this function is equal to zero. Essentially, this's saying that when rules generous, satisfied, we know for a fact like one hundred percent sure that there's at least one C one number one point along the X axis on the function where the derivative is equal zero. So let's look for the derivative. Well, we see one raise your exit goes to because we can draw a straight line. I hear well, that's not very straight. But you see my point. This is a straight line that X equals two. So we have one at X, he called two. And if we look, actually, it's also at X equals five. We have another, another great line. Another tangent such that the slope, the slope Halftime of C is equal to zero. So those are the two point in which the those are the two valleys of seat that satisfied throws.