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Problem 60

a. Find the local extrema of each function on the…

Problem 59

a. Find the local extrema of each function on the given interval,
and say where they occur.
b. Graph the function and its derivative together. Comment on
the behavior of $f$ in relation to the signs and values of $f^{\prime}. $
f(x)=\csc ^{2} x-2 \cot x, \quad 0<x<\pi


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Video Transcript

Okay, so here you have that Dex is negative to cosign. X minus cosign square. Dex from the domain is negative by Tau pi. So if we take the derivative to find our critical points, this is going to be to sign X and this is going to be plus two sigh Next cosign x. It's a bit of coastline squared is to cause the next times negative sign X. So that makes us a plus. Okay, the derivatives always defined. So let's look at when have prime zero. Well, that's gonna be when Okay, seiken factor out. I'm just saying this to be zero. So if I factor out to sign axe and then I have one plus because I next that's just f crime. When is that zero. What's when sine X is equal to zero? So that happens at the end points of X equaling negative pie and pie. And then when is this equal to zero? And it's when coastline X is equal to one well again. That happens when X is equal to either plus reminds Plaster accrue points truly are just plus or minus pi, So if we plant these on a number line and look at the derivative. No, thank you, Sky High. We just want to know the value of the derivative inside here. So let's just play in zero. That's easy enough. We're going to get time. So there's actually I'm sorry. There's one more critical point win. Consign xB. Zero also at X equals zero to get us down. Zero. Here we go. Let's look att f crime. So between negative find zero it's blocking negative pyre or two that's going to give me a negative too. And then Plus, this is going to be zero that's going negative. And then here we'LL have a pirate too. Plus zero Ah, sorry, Sinan Pie over two times to which is two plus zero, which is positive. And so we have a local men at well, at zero and what's the value? It's gonna be negative three slick and then we have local max at the two m points because we're decreasing and then increasing. Okay, so at closer minus pi, they actually have the same value. So this is going to be plus two minus one. So one so local max of one of plus or minus pi, and again if we look at the derivative graph of the function, we should see the coordination between positive derivative increasing function, negative derivative, decreasing function.

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