Mean Value Theorem Help?

Question

i need help finding the secant line from ( a, f(a) ) to ( b, f(b) ) and then the tangent line that is parallel to the secant line.
f(x) = x + (1/x) [ .5, 2 ]

Answer 1

f(1/2) = 5/2 and f(2) = 5/2. The secant line goes fron (1/2,5/2) to (2,5/2), so the slope of the secant line is 0. Now
f'(x) = 1 - 1/x^2 = (x^2 - 1)/x^2. This has slope 0 at x = 1, so you want the tangent at the point (1,2).

Notice that the derivative also vanishes at x = -1, but that is not in the interval [1/2,2].