Calculus homework help

Question

The mean volume theorem states that if f is continuous on [a,b] and differentiable on (a,b) then there is some number c £ (a,b) such that:

f’ (c) = f(b) - f(a) / b-a

for f(x)= 2x³ +7, find such number C in (1,3)

Answer 1

First we will calculate the mean rate of change over the interval.

f(3) = 2*3^3 +7 = 61

f(1) = 2*1^3 +7 = 9

f(3)-f(1) / (3-1) = 26

Now we must solve the equation f'(c) = 26

f'(x) = 6x^2 from the power rule so

6c^2 = 26

c^2 = 13/3

c = sqrt(13/3)

and sqrt(13/3) = 2.08... so it is indeed in our interval