Problem

Let $f(x)=1-x^{2 / 3}$ . Show that $f(-1)=f(1)$ b…

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Problem 4 Medium Difficulty

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers $c$ that satisfy the conclusion of Rolle's Theorem.
$f(x)=\cos 2 x, \quad[\pi / 8,7 \pi / 8]$

Answer

$c=\frac{\pi}{2}$

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Calculus 1 / AB

Essential Calculus Early Transcendentals

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 2

The Mean Value Theorem

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Video Transcript

the function f of X people to co sign of two X found in trouble from hi over 8 to 7. Pi over. It is continuous and defensible since a co signer sine function, it is continuous indefensible for a real values of X. So this function says rise the 1st 2 hypotheses of girls there. So we need to check the third hypotheses hypothesis worlds there and which requires that the function has equal value. That's the point of internal so. But those first track the function at X equals pi over eight. So you have co Sion who two times pi over eight, which is the Coast Zion are high over four, which is squared to over to and that struck the function at seven high over eight. Some have close eye and no two times seven pi over eight, which is seven pi over four, which mod to pie is also a cold mountain negative high over for which will have for the same value squared up to over two year. So then let's find any values between pirate and seven pi over it that will then have a slope of zero. Since this will views you to tell the coastline function will be rising and pulling in between. Um, this is either and it ah, hole period or half a period. So then let's find for the drivel function. So it'll be minus two times sine of T rex South ID equals here. So then we heard so and very sign of zero is equal to backs. The sine function is your at Elia's zero high Tu Po, etcetera and also negative ones. But we can also already tell that we need something that's positive equal to two X. So then we would under put backs equals zero or pi over two. Hi, and the only one that is between higher eight and some Empire eight here is high over two. So then we end up with a C Valium of high over to

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