Answer:
(a) The average value of the given function is 12/π
(b) c = 1.238 or 2.808
Step-by-step explanation:
The average value of a function on a given interval [a, b] is given as
f(c) = (1/(b - a))∫f(x)dx;
from x = b to a
Now, given the function
f(x) = 6sin(x) - 3sin(2x), on [0, π]
The average value of the function is
1/(π-0) ∫(6sinx - 3sin2x)dx
from x = 0 to π
= (1/π) [-6cosx + (3/2)cos2x]
from 0 to π
= (1/π) [-6cosπ + (3/2)cos 2π - (-6cos0 + (3/2)cos0)]
= (1/π)(6 + (3/2) - (-6 + 3/2) )
= (1/π)(12) = 12/π
f(c) = 12/π
b) if f_(ave) = f(c), then
6sinx - 3sin2x = 12/π
2sinx - sin2x = 4/π
But sin2x = 2sinxcosx, so
2sinx - 2sinxcosx = 4/π
sinx - sinxcosx = 2/π
sinx(1 - cosx) = 2/π
This equation can only be estimated to be x = 1.238 or 2.808
