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Problem 61

The graph of $f^{\prime}$ is given. Assume that $…

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Problem 60

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}. $
$$
f(x)=\sec ^{2} x-2 \tan x, \quad \frac{-\pi}{2}< x <\frac{\pi}{2}
$$

Answer

See Graph


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Video Transcript

All right, So we have ethics equal seeking squared Tex Minds too. Tan X. And this is between plus and minus pirate too. Both of these functions are going to be defined in the Centre Bowl and surf Addictive derivative. We'LL get two and then seek an axe And then times the derivative of sea connects that seeking next tannic. So we'LL get a Sikh and squared X tan axe and then minus too derivative Tan X is seeking Squared X Okay, so when there's a crime equals zero will weaken factor out too seeking square decks and we're left with tan X minus one You know that to be zero should know By now seeing and squared is never equal to zero That's always greater than one in less than minus one. Less cynical to create a magical too. So we wantto windows tan x equal one. In this interval there's only one point and that's that pirate for atnegative hire four tens Negative one secure the only critical point. And so let's plot it along with her ten points that aren't included Really eerie apart for Okay, So if we look att se zero we're going to get one writes a seeking squared of zeros, one to be positive. And then if we look at say, pie or three, this's going to be I'LL see for thirds and then three A little tricky Um oh, you know, I'm looking in the wrong things. Look in Europe it anyway. It's a derivative. Yes, if applicants here to the derivative I'm actually going to get, uh, let's see. I mean, get one. I mean, times zero minus ones is actually negative. If I plug in zero sequence created. Zero is one ten zero is zero and then minus ones is negative. And then, as a good cross across pirate before no seek, it is going to stay positive things. Question is always gonna be greater than zero. There's anything about Tan X and Tan X when X gets greater than fire for Tan X will be crater than one, so this term will be positive. So have a positive. And so it looks like I have a local men I can't hi ever, for with the value of thank you, it's going to be two minus two, says Europe and no Uncle Max. That's because the end points you're not include the function is not to find her. Okay. And again, if we look at the graph, we should see this coordination between the derivative being positive and the function increasing and the driven a big negative on the function to reason.

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