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DEFINITIONS Let f be a real function defined On an interval [. Then fis increasing on if x < y implies f(x) fly) for allx, yin I And, fis decreasing on I if x &l...

Question

DEFINITIONS Let f be a real function defined On an interval [. Then fis increasing on if x < y implies f(x) fly) for allx, yin I And, fis decreasing on I if x < Y implies f(x) > fy) for all x, y in [.

DEFINITIONS Let f be a real function defined On an interval [. Then fis increasing on if x < y implies f(x) fly) for allx, yin I And, fis decreasing on I if x < Y implies f(x) > fy) for all x, y in [.



Answers

A function value $f(a)$ is a relative _____ of $f$ if there exists an interval $\left(x_{1}, x_{2}\right)$ containing $a$ such that $x_{1}<x<x_{2}$ implies $f(a) \geq f(x)$.

Here's a visual to help us understand this problem. So if we have an interval where when X sub one is less than except to like we see here, f of X of one is greater than f of X, up to like we see here, we can see this point has a higher Y coordinate in this point, and that's true all along the points on the interval. If that's the case, then this is called a decreasing function, or that least on the interval. Dysfunction is decreasing.

Here's a visual to help us understand what this problem is saying. So we have an interval from X, someone to accept to, and we have some X value that's being called a That's in that interval. And so if we know that f of A, which would be the Y, coordinate of the point is greater than or equal to the Y coordinate of any of the other points in the interval. So then the other ones are lower, whatever that happens to look like they're lower or their equal to it. If that's the case, then what do we call this point or what do we call that value that why value? We call it the maximum?

Okay, so I want to show it that if f opec's is increasing is increasing, then this is only if early here g as a function of experts to define as one over it has a function off X is strictly decreasing. So now it will go in this direction. So we assume assume f as function o X is increasing. So what does it mean to be increasing? Well, it means that if X is less than why then as a function o x, it was function Oh, exes Lesson at as a function. Why now? Tape taking reciprocal Czar, Both sides. This implies that one over a za function off X because you're taking reciprocal is you have to fit the sign. So this is one over at as a function of why so now if you define geo exited final part isn't that g of X is greater than G A y. Um when x is less than why so Therefore you have g off X is decreasing. Now you have to prove the opposite way. So we have to go this way. So if G o X is equal to one over X is decreasing well then once again that when X is granted one, then G as a function off X must be less than G as a function off. Why so substituting as one of our April banks that said one over at as a function of why so taking reciprocal was once again you have to fit the signs away. So therefore, if oh X is increasing So since you since you've proved both both sides of the implication that if ever begs, is increasing in This implies that g o X is decreasing. So this was proved on page on the first page. Now you also prove the backwards one. So you proved that when G of X is decreasing So when this is decreasing in their four best is increasing So therefore you have every base is increasing even any if genomex is decreasing

Okay, so in this question, we want to prove that effort back. That's strictly decreasing even. And if g o x, which is one over for Beckett strictly increasing. So because this is an if and an if question, we have to prove the 1st 4 direction on the backwards direction. So we just prove the four directions first. So let's f o X being street. We did so this stand by definition, if X is greater than why look. So if X is less than why in this implies that f all ex is going to be rather than f or why this is just a very simple definition off the losing. So what does this mean? If we flick both of these sides, then you have one over F or X is greater than one over off took notice. Um, switch like this. Now we have a deal. It's one of the place. So this implies that g o X is less than a genius. Why? So what does this mean then? So this means that if X is less in why this implies that G o X is less in G off Why? Which implies that G off X is strictly in prison by definition. So therefore we've proven the first direction. So to prove the backwards direction let g o X, which is one over X the strict Wait, increasing. So what does it mean to be increasing? It means that if X is less than why and supplies that g o X is lesson g o Why? Well, what is your next G of X? Is just one over f o X just doesn't one ever f why Now we've heard everything. So we have to sign as well. The effort X is then granted an f Why What does this mean? What does this mean? That So it means that if X is less than why he implies if X is greater than ever why? Which implies that if he's strictly defense as a result, we've proven the backwards direction. So therefore the statement Fuller's


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