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

the mistake in the pattern is that :

b. the decimal point was moved one too many place values each time

the correct pattern could be made by :

b. moving the decimal point to the right the same numbers of place values as the exponents