Verify that the function f(x) = x+ sin(πx) on [1,2] satisfies the hypotheses of the Mean ValueTheorem on the given interval.

Question

Then find all real numbers which satisfy the conclusion of the theorem. Give the exact value of the numbers, not a decimal approximation.

Answer 1

it is a continuous function....

f(1) = 1 + sin(pi) = 1 + 0 = 1

f(2) = 1 + sin(2 * pi) = 2 + 0 = 2

so the slope of the secant line is (2-1)/(2-1) = 1/1 = 1

f'(x) = 1 + pi* cos(pi*x)

1 + pi* cos(pi*x) = 1

pi*cos(pi*x) = 0

cos(pi*x) = 0

cosine is zero when the angle is 90 degrees=pi/2 or 270=3*pi/2

if the angle is pi/2 then pi*x = pi/2 --> x = 1/4 which is not in the interval

if the angle is 3*pi/2 then pi*x = 3*pi/2 ---> x = 3/2 which IS in the interval

So when x=3/2 , f(3/2) = 1/2

f'(3/2) = 1 which agrees with the slope of the secant line