Problem: A painting is hung so that the top is b feet above your eye level while the bottom is a feet above your eye level. Let x be the distance between you and the wall that holds the painting. Your goal is to stand in a place such that the angle Q subtended by the painting at your eye (i.e. the angle between the top and the bottom of the painting with respect to your eye) is as large as possible.

1. Express Q as a function of x.

2. Determine the values of x for which Q is increasing and the values of x for which Q is decreasing.

3. Use your answer to the above to determine where you should stand.

Answer 1.

Q = arctan(b/x) - arctan(a/x)

Answer 2.

Q ' = -a / (x^2*(1+a^2/x^2)) + b / (x^2*(1+b^2/x^2))

Now that I have the derivative of Q, I think I need to find values of x where Q' > 0, Q' < 0

I'm having trouble simplifying the equation.

-a / (x^2*(1+a^2/x^2)) + b / (x^2*(1+b^2/x^2)) < 0, for x

-a / (x^2*(1+a^2/x^2)) + b / (x^2*(1+b^2/x^2)) > 0, for x

Any suggestions?