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If $f(x)=x^{2} quad$ is an increasing function, then (here $a>0$ )(a) $x in[a, 2 a)$(b) $x in(-infty,-a] cup[0, a]$(c) $x in(-a, 0)$(d) None of these...

Question

If $f(x)=x^{2} quad$ is an increasing function, then (here $a>0$ )(a) $x in[a, 2 a)$(b) $x in(-infty,-a] cup[0, a]$(c) $x in(-a, 0)$(d) None of these

If $f(x)=x^{2} quad$ is an increasing function, then (here $a>0$ ) (a) $x in[a, 2 a)$ (b) $x in(-infty,-a] cup[0, a]$ (c) $x in(-a, 0)$ (d) None of these



Answers

Order the following functions from slowest growing to fastest growing as $x \rightarrow \infty$ .
a. $2^{x} \quad$ b. $x^{2}$
c. $(\ln 2)^{x} \quad$ d. $e^{x}$

This question is a vehicle of choice that says which of the following If the range of F of X equals one plus one over X minus one. So what we can do is we can actually just determine the domain range is good practice. So the domain is going to be pretty much anything except for when X is one. So we can say that we can have negative infinity. We can't have one, but we can't have everything above one as well. But then the range is going to be all of the Y values that exist when we have this domain, so we can think about it. If X is really, really, really small, really negative, we're going to have a number that's slightly under one. But it's just going to be represented as a negative infinity. And then we're never actually going to have one, because having one would require this fracture to be zero, which could never happen. So we're gonna have a parentheses there. But then, if X is positive, we again can't be won. But we can be any number above one, so we'll have infinity here and looking at our answer choices, it's represented the range represented quickly and answer choice, eh?

For 45. We're looking for the right hand derivative of F S. O. For That's to be the case. Um, we're gonna look at the bottom because we're looking at the right hand derivative of X equals zero. So that means we're gonna end up with the limits as h approaches zero from the right, uh, of zero plus h minus up of zero all over h. And if we go ahead and plug that in, we end up with just a JJ squared, plus one minus one over h. And if we then simplify this, we can reduce the ones the H squared over age can be reduced A cz well to just h. And if I plug in the limit, zero h, I get zero, which is answer C.

You know, in order to see which function grows faster or slower than the other, we're gonna have to compare them. So first, if we look at the functions that have underlined in red, where looking at X squared and the natural other two to the X power. So if we live with the lemon is that's a purchase and Vinny of Alan to to the act's over X squared, we can evaluate this limit using lobby college rule. So one year's flop casual gives us the natural log of the natural of of two times Allen of Teach the X over two times acts using Levy cultural one more time we'll let us evaluate this limit. So we're going to get a gallon of Alan of two squared times, Ellen up to to be ex all over too well, this is really just a constant Ellen of Alan of two squared over two times. The limit is extra. Purchase infinity of the natural out of two to the ex. Since the natural log of two is less than one, that limit is zero, which means that the natural log of two to the ex grow slower than acts where next we'll compare the functions I've underlined and blue and then the ones of underlying ingrate. So for comparing X squared and two to the X, then the limit is that's a purchase infinity of X squared over tune of the X We can evaluate using lobby tolerable again. One use of lab casual gives us the limit is except artist vanity of two acts over Eleanor, two times two to the ex. One more use of lobby cultural gives us a limit is Axel purchase infinity of two over the natural log of two squared times two to the ex. This limit is zero, which means that X squared grows slower than to to the ex. So now let's compare E to the X and two to the X to the limit as that's a purchase. Infinity of each of the AKs over to to the ex is the limit is X approaches infinity of eat over to to the ex, since he's greater than to see that in over two is granted them one, which means that this limit is infinity. So that means that each of the experts faster than to to the ex, so putting this all in order. The function that grows from slowest two fastest in that order is Alan or two to the ex. The next one is X squared. The next one is to the acts and finally need to the ex grows the fastest.

Hello, everyone. I hope all is well. Today I will be helping you with the eighth problem of the model for test. And the 8th 1 is asking if FF X is greater than or equal to zero for all of X and then s o then f of two minus acts is what Brent? So we're talking about f of X, right? We're talking about the range, so the range is either greater than or equal to zero. So then it would have, um, the range again. So it's asking how will the range react when you have two minus x? Correct. So the F of two minus acts had just shifts and reflects the graph horizontally, and it does not have any vertical vertical effect on the graph. So regardless of what substituted for X, f of the range will not change. So, for example, if we had a parabola, essentially all that would be affected correct would be the horizontal shift. So it might be over here, or it could be over here, etcetera. But the range would not change. So for that reason, it would be be it would still be greater than or equal to zero. So I hope you found the cell phone and I'll be over. Great day. Thank you.


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