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### Discuss the extreme-value behavior of the functio…

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Problem 68

Sketch the graph of a continuous function $y=h(x)$ such that
$$\begin{array}{l}{\text { a. } h(0)=0,-2 \leq h(x) \leq 2 \text { for all } x, h^{\prime}(x) \rightarrow \infty \text { as } x \rightarrow 0^{-}} \\ {\text { and } h^{\prime}(x) \rightarrow \infty \text { as } x \rightarrow 0^{+}} \\ {\text { b. } h(0)=0,-2 \leq h(x) \leq 0 \text { for all } x, h^{\prime}(x) \rightarrow \infty \text { as } x \rightarrow 0^{-}} \\ {\quad \text { and } h^{\prime}(x) \rightarrow-\infty \text { as } x \rightarrow 0^{+} \text { . }}\end{array}$$

See graphs.

## Discussion

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## Video Transcript

All right. So we're sketching the graph of a genius function. It looks like h is between negative two and two. It's important. There's continuous. So that won't have any pasin toads or anything. And then Rios, same thing h of zero suit. So let's do And red. So h of zero is zero. Oh, wait, I see. Um and I was not remain is between negative to do. It's at the range is between negative Teo So h of zero is zero. Yes, the fuck said this condition is saying that the function has to live between negative tio one two So h of zero zero, we got that h prime avec ce approaches. Infinity is experts zero from the left. Okay, so that means we're approaching a vertical tangent at once. And from the left, we have a vertical tangent going up this vertical. And also just to the right. We have a vertical tangent. And then I think that's the only condition we need. Okay, Good. And so your b h of zero is still zero this time H is between zero and negative two And that ch crimes approaching infinity from the left. So we're going to have a vertical tangent like this, but it's approaching minus infinity from the right. So it's going to be down on negative like that was going to be a customer. Okay, so we have a vertical tenant here and here's a customer.