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Numerade Educator

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Problem 27 Easy Difficulty

Does there exist a function $ f $ such that $ f(0) = - 1 $, $ f(2) = 4 $, and $ f'(x) \leqslant 2 $ for all $ x $?

Answer

Suppose that such a function $f$ exists. By the Mean Value Theorem there is a number $0<c<2$ with
$f^{\prime}(c)=\frac{f(2)-f(0)}{2-0}=\frac{5}{2},$ But this is impossible since $f^{\prime}(x) \leq 2<\frac{5}{2}$ for all $x,$ so no such function can exist.

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Video Transcript

The question here is asking that there exists a function after Stan effort. Zero is equal to negative one and Africa Tuesday quota four and five to dig for an F Prime Max is less than equal to two for all act. So here we can kind of think that at a mean, that is a promise. So we are in this interval. Zero Closing around from door to two is what we will assume, and it's asking if there is a such a nap time. Iraq's this less than equal to. So why don't we go ahead and solve for Why don't we just go and imply that means I erm which is going to be effort too Minus zero Oliver two minus. You know, since we have those two values, the average slope will get give a for minor negative one over two, and the negative and the negative makes a positive. So this will be four plus one this kiss bye over to you and said, this is the slope. So now what does this mean? What does this exactly mean? This means this value five have told us that at some point along along zero long zero and four, the slope is going to equal five half. That's what that's what. That's what. This That's what this is telling us that on some function from zero to two on any basically on any function that is on a closed interval, there is some point. At some point along this this access there's going to be a C that will give us a slope of five. And since five have is greater than two, thiss is not true. This is this is no no, There's no function that will always give us his flute lesson equal to because we just proved that the average slow is five half. It is greater than two. So this was saying that this is for all for all actual we just clearly proved that there is one c one x that has a slope greater than two. So this is not true. So not true. No,