Find the value or values of c that satisfy the Mean Value Theorem

Question

f(x) = x +12/x, [3,4]

Answer 1

Find f'(c) that satisfies Mean Value Theorem:

f'(c) = (f(b)-f(a))/ (b-a)

= (f(4)-f(3)) / (4-3)

= [(4 + 12/4) - (3 + 12/3)] / (4 - 3)

= (0) / (1) = 0

Find f'(x):

f'(x) = d/dx[ x + 12/x ] = 1 - 12x^-2

Solve f'(x) = f'(c) to find values of c that satisfy the Mean Value Theorem:

1 - 12x^-2 = 0

1 = 12x^-2

x² = 12

x = +√12, -√12 = +2√3, -2√3

So, c = +2√3, -2√3 satisfy the Mean Value Theorem (let me know if there are errors or it is unclear)