Question

Find the average of the function over the given interval and all values of x in the interval for which the function equals its average value. (Round your answer to three decimal places.)
f(x)=4 cos(x), [0,pi/6]
(x,y)=(___?___)

Answer 1

the avg value is:
\[\frac{1}{b-a} \int\limits_a^b f(x)\]
\[\rightarrow \frac{6}{\pi}\int\limits _0 ^{\pi/6} 4\cos x dx\]

Answer 2

Not sure where to go from there

Answer 3

compute the integral, which is not too hard because the anti derivative of \(\cos(x)\) is \(\sin(x)\)

Answer 4

you have
\[\frac{24}{\pi}\sin(\frac{\pi}{6})\]
\[=\frac{24}{\pi}\times \frac{1}{2}=\frac{12}{\pi}\]

Answer 5

not sure how you are going to solv e
\[4\cos(x)=\frac{12}{\pi}\] i guess divide by 4 and get
\[\cos(x)=\frac{3}{\pi}\] so
\[x=\cos^{-1}\left(\frac{3}{\pi}\right)\]