How to find x values in interval

Question

_{The question says: Find the average value of the function over the given interval and all values of x in the interval for which the function equals its average value.} 

 

I found that the average value is 2/Π but I don't know how to find the 2 x values. 

 

f(x)= sinx [0,Π]

 

Thanks! :D

Answer 1

Hi Rebecca,

 

You found the average value of the function on the interval from 0,pi to be 1/pi ∫₀^πsinx dx = 2/pi. Now what values of x in the interval from 0 to pi generate a function value equal to 2/pi? That is, when does sin x = 2/pi? To solve that equation for x use the inverse sine (or arcsin) function, i.e. x = arcsin (2/pi). Your calculator reveals that to be approximately 0.6901 radians. There is one more x value in the interval that produces the same result (pi - 0.6901) or approximately 2.4515 radians.

Answer 2

f(π) = sin(π) = 0

 

f(0) = 0

 

 

Since both f(x) values are zero, the average must be zero.