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Sketch the graph of a function that satisfies all of the given conditions

$ f'(0) = f'(4) = 0 $, $ f'(x) = 1 $ if $ x < -1 $,

$ f'(x) > 0 $ if $ 0 < x < 2 $,

$ f'(x) < 0 $ if $ -1 < x < 0 $ or $ 2 < x < 4 $ or $ x > 4 $,

$ \displaystyle \lim_{x\to\2^-} f'(x) = \infty $, $ \displaystyle \lim_{x\to\2^+} f'(x) = -\infty $,

$ f"(x) > 0 $ if $ -1 < x 2 $ or $ 2 < x < 4 $,

$ f"(x) < 0 $ if $ x > 4 $

The function is increasing with a slop of 1 for $x < -1 .-1$ is a local max because the first derivative changes sign. For $-14$ the function is concave downward. 4 is ruled out as either a max or min because the 1 st derivative has the same sign on both sides.

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all right, so we're being asked to stretch to graph a function that satisfies all of the given conditions. And so we obviously have a lot of conditions. So I'm going to go ahead and kind of put this all together in a formal intervals. Easy the way to understand. So let's see. Let's start off with this stuff down here all the stuff. That's a lot of interval stuff. So we're goingto see. So says the derivative So at front of X is greater than zero if zero of Weston access Weston to. So that means that if it's positive the function the graph is increasing, so is increasing from zero two two and then we were told that it is less Inger between is decreasing and this is occurring from negative one to zero. And then this is also occurring between ah, all ex Celestine too. So this is going to be all lessons all ex, lest into and less than four. So between two and four and and all greater than force often for to infinity. And then we're told that the limit as experts is negative to me. Two from the left is to part of infinity and from the right it's negative Infinity! So we have a vertical ascent. Oh, at two. Ex tickles too. And then we're told that the Khan cavities. So we're told that in this car that from negative one to two and between two and four, it is also concave down for all ex greater than force that I mean for to infinity. So now that we have all of it information we can could've graft this picture now. And we're also told that the at thes access we have a tangent. We have ah, attendant line of C e o. And we're also told that FBI director, I missed this information at one. It is equal to one, so yeah. Ah, let's go ahead and start drafting. Um Okay. So are critical for him to start from the left side. We're going to go from the left to the right. We know that it is increasing between zero and two and we notice decreasing between for ah, for all ex lesson zero and yeah, so this is gonna look something like this and make a new one kind of kind of come up. You can have a zero. Is gonna have a a a local, Max, It's going to come down. We're gonna have a tenant line of zero and then at Toothless, just labelled with that too. This is going to go after positive infinity. And this is going to come down since Meghan Infinity. So we're gonna have it increasing, and they're gonna have a changing con cabbie right here in the local max and then come down and this is happening at force, and this is a graph.