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For what values of $c$ is the function$$f(x)=c x+\frac{1}{x^{2}+3}$$increasing on $(-\infty, \infty) ?$

For all $c>\frac{1}{8} \quad f(x)$ is increasing on $(-\infty,+\infty)$

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 3

Derivatives and the Shapes of Graphs

Derivatives

Differentiation

Applications of the Derivative

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in this question. We want to figure out what's this condition for? C can make this ffx to be increasing from negative infinity toe positive infinity. So let's take the first of the volunteer for it first. This is the first of the narrative. So if if if, If, yes, increasing everywhere. That means if prying off x yes, uh, districted is positive or known active everywhere. So we look at this first of the directive, the numerator. So this denominator is always, always positive, Um, use We need to, um, make the numerator to be positive. So we defined. So that means we what? We want it. All we need is to solve this equality. See, times X square past three square minus two x is greater than zero I will wear so we can rewrite this into see greater than two x over x square past three square. I just moved these two X uh, to the right and the divided by the cultivations X square plus the race square. So have this conclusion IFC satisfy these can condition. We have, uh we have a increasing function. If no, we look at this conclusion here. So we let with defined The new function Geo FX equals to this two x over X square past the race square on the idea is to find the maximal value but you So if we can do this, if we can find the next mix maximum value for G. So if C is greater than this Mexico value off cheap, then we have this condition. That means FAA is increasing everywhere. So we basically can wrote this program to this A noose of seven program. Well, to find out a maximum very effort you. So in order to do this, we can just take the first of the narrative of Jeep. So we have. If we do that, be careful about this. It's ah, it's about the It's about the question. Ruin the train rule. So the result should be, um, six miners, six x square, divided by X square, past three square. So wait, you We can check this, but this is the first of the purity of off. So we just let g prime equals zero. So we have two solutions. X equals to one X equals minus one. That means we need to analyze the three intervals from minus infinity to minus one from minus 1 to 1 from one to infinity. So for the first interval that G prime the first of the curative off, it's actually our intimacy. So it's actually negative. So it's decreasing and from minus went toe one is positive. So seeing crazy ing and from one to infinity, it's an active. So it's decreasing so indeed that we have a local makes. Among I had two x equals to one with value. If one saris g for very G one equals Teoh, Uh, we can plug in X equals wanted you. So we have one over, um when over eight. But can we argue that this is the global mental? Because it's a local maxim, which is no enough. We need a global maximum for G. So this is the increasing decreasing Terrible. But we also have some, um a simple, totally behavior. Fergie. So no catchy off X here he has to horizontal a sympathetic If we send ex coast to positive infinity or next infinity, we have zero. So in fact, we can graft this G gee effects on our coordinate. So you really looks like this. Say you decreasing for us now we touch, um, minus X equals two minus one. It becomes increasing and the crazy on period touch pass X equals to one. It's decreasing here. So this point is not only a local mentum, and it's also a global maximum, which equals to one. All right, so our conclusion is, if C is greater than one of rate GOP, um, you've sees great and one rate that f prime of X is positive everywhere. That means effort is increasing everywhere. So this is our results C squared and wondering.

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