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AK
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Problem 24 Medium Difficulty

Sketch the graph of a function $ g $ for which
$ g(0) = g(2) = g(4) = 0 $, $ g'(1) = g'(3) = 0 $, $ g'(0) = g'(4) = 1 $, $ g'(2) = -1 $, $ \displaystyle \lim_{x \to \infty} g(x) = \infty $, and $ \displaystyle \lim_{x \to -\infty} g(x) = -\infty $.

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Video Transcript

Let's create this function G of zero is equal to GF two Is equal to GF four which is zero. Okay, so what does that mean? That means at the X values of 402 and four, 0, 2 and four. We have Why values of syrup. Okay, G prime of one Jeep. I'm going to write this one down. G prime of one Equals G. Prime of three. Keep Promise three. Oh okay. Mhm G prime of three which is equal to zero. What does that mean? Well, G prime means slope, that's what your prime means. So that means at one at the x values of one and 0 at the X values of one and zero. We're going to have like a flat line basically where the derivative is neither increasing nor decreasing. Its Slope is equal to zero. That's what that means. Okay, G prime of zero Is equal to G prime of four at positive one. Okay, so that means the slope at these places is positive one. Yeah, Positive one. Kind of like that. Not exactly, but kind of like that. Okay, what else? G prime of two is negative one. So the slope is going to be negative one there. Okay, what next? As X approaches infinity? The limit goes to positive infinity. Okay, that means as x increases this way this function is going to go up towards positive infinity. Okay, and the limit as X approaches negative infinity. So on the left here is going to be negative infinity negative infinity. So let's connect all these dots here we have our zero slopes. our function is going to look something like this, it's gonna come up, hit zero, it's going to level out and I'm sorry this is a little wavy, should be straight having some pent issues Then it's going to be negative one It's going to come up again. So at zero. So it x equals three. It has a zero slope. Oh, why didn't I think of that? It doesn't even have to come up right. It can have a zero slope down here and then it's going to have a negative one. A positive one slope through there. So this function can look something like this. You can have some more waves on the outside But at this fulfills all our conditions, G of zero equals G f two equals G f four equals zero. So these are all zero points at G prime of one. G prime of one equals G prime of three equals zero. So that means that this equation, the slope is going to be zero at these points which the scrap fulfills and G prime of zero equals two. Prima four as positive one, yep. The slopes of these functions here are about positive one and then do you promise to equals negative one? A negative one slope at X equals two limit as X approaches infinity as positive infinity, yep. See it heads up this way like that and then as X goes to negative infinity, we also get negative infinity On this side. This is our negative infinity. This is our positive infinity, our X values. So we had to negative infinity as X approaches negative infinity and positive infinity as X approaches infinity. So that is how you graph a function that fulfills all of these conditions.