Question

Find the values of c that satisfy the equation f(b)-f(a)/b-a= f prime(c) in the conclusion of the mean value theorem for the given function and interval. f(x)=9x+9/x [1/9, 9]

Answer 1

Due to having to type, my arithmetic may be off, but the steps should be clear.

First solve for f(a) and f(b)

f(a) = 9a + 9/a = 9(1/9 + 1)/(1/9) = 90

f(b) = 9b + 9/b = 9(9 + 1)/(9) = 10

Then solve for f'(c) = f(b) - f(a)/(b - a)
= 10 - 90/(9 - (1/9) = -90

Then differentiate f(x):

f'(x) = 9 - 9/x^2

Substitute c:

f'(c) = 9 - 9/c^2 = -90

Solve the equation for c!

I haven't checked for errors, but the steps should be clear.

Answer 2

I am not getting 90 or 10...

Answer 3

My window closed while answering.

First solve f(b) then solve f(a) then solve b - a:

Since f(x) = 9x + 9/x,

f(a) = 9a + 9/a = 9(1/9) = 9/(1/9) = 1 + 9 = 10

Do the same for f(b).

b - a would be 9 - 1/9

Then f'(c) would equal f(b) - f(a)/(b - a)

Then find f'(x).

f(x) = 9x + 9/x leads to
f'(x) = 9 - 9/x^2 (check this)

substitute c:

f'(c) = 9 - 9/c^2, which equals f(b) - f(b)/(b - a)

That should give you an equation for which you can then solve for c.

Answer 4

9/(1/9)=81

Answer 5

\[f(1/9)= 9(1/9)+\frac{ 9 }{ 1/9 }\]