Question

Find the critical numbers of the function.

Answer 1

\[g(\theta)=12\theta-3\tan \theta\]

Answer 2

solve
\[12-3\sec^2(x)=0\]

Answer 3

g'(x)=12-3(secx)^2=0
-3(secx)^2=-12, (secx)^2=1/4, secx=1/2, secx=-1/2

Answer 4

cosx=2, cosx=-2, no critical number

Answer 5

lets see if it was cooked up to be nice:
\[3\sec^2(\theta)=12\]
\[sec^2(\theta)=\frac{12}{3}=4\]
\[\sec(\theta)=\pm2\]
\[\cos(\theta)=\pm\frac{1}{2}\] yup, cooked up nicely

Answer 6

So the answer is 2,-2,1/2,-1/2?

Answer 7

@kittyxie1 \(\frac{12}{3}=4\) rather than \(\frac{1}{4}\) i think you took the reciprocal anticipating that you were going to have to do it later to change secant in to cosine

Answer 8

no the answer is in terms of \(\theta\) not \(\cos(\theta)\)
what is \(\theta\) if \(\cos(\theta)=\frac{1}{2}\) ?

Answer 9

\[n \pi/3\]

Answer 10

?

Answer 11

\(\frac{\pi}{3}\) is correct

Answer 12

then what \(\theta\) if \(\cos(\theta)=-\frac{1}{2}\) ?

Answer 13

oh i see, it is not \(\frac{n\pi}{3}\) it is
\[\frac{\pi}{3}+2n\pi, \frac{2\pi}{3}+n\pi\] for the first one