Consider the given function and the given interval. f (x) = 6 sin (x) - 3 sin (2x), [0, π]

(a) Find the average value fave of f on the given interval.

(b) Find c such that fave = f (c). (Round your answers to three decimal places.)

(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