Question

Graph the function f(x) = x + 4 / x

Graph the secant line that passes through the points (1,5) and (8,8.5) on the same set of axes

Find the number c that satisfies the conclusion of the Mean Value Theorem for f on [1,8]

Answer 1

So do I have to find the second derivative, and critical points? then sketch it, THEN find the line? or i can just use f'(c) = f(a)-f(b)/a-b ?

Answer 2

Some hints from algebra would be that there is a vertical asymptote at x=0 and a slant asymptote at y=x.
A first derivative test will then tell you the extrema. At that point a second derivative test will just confirm what should be obvious at that point, so I would just graph it without the second derivative.
You can draw the secant line by simply drawing a line through the given points on the graph. They didn't ask for the line's equation, so I wouldn't look for it.
Then use your Mean Value Theorem as you stated to find the x location of the tangent line whose slope matches the slope of the secant line.
They didn't ask, but if you draw a line on the graph of the function with that slope at the x location that you find, you should have two parallel lines on your graph.