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Problem 65

Sketch the graph of a differentiable function $y=…

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Problem 64

Show that the functions have local extreme values at the given values of $\theta,$ and say which kind of local extreme the function has.
$$
h(\theta)=5 \sin \frac{\theta}{2}, \quad 0 \leq \theta \leq \pi, \quad \text { at } \theta=0 \text { and } \theta=\pi
$$

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Video Transcript

Okay, so we have dysfunction h of data. It is five times signed data over to and we would like to show that dysfunction is a local extreme at ADA equals zero Clint data equals pi, and data is between zero and pi scared. So we'll show. Actually, Is that the driven in the first ball of the derivative? It's gonna be five house co sign of fate over to so notice that h prime zero is going to be one teenage crime of jai is going to be zero, and some pie actually is a critical point. But it's okay that zeros on a critical point because it's the end point o r function, so zero and pie And I actually have no critical points between zero and pi. So then in free, just plug and say fire two, we're going to get co sign a pi over four, which is gonna be positive. It's actually her function is always increasing between zero and pi. So we're gonna have a local max at hi, and the value is actually going to be so since sign of poverty is one. It's going to be five, and we have a local men at here and the values going easier because sign of zero zero. And these will also be absolute local man, plus absolute, because the function is strictly increasing between zero and by. So the lowest point is going to be zero, and the highest point is going to be pi X equals Y.

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