optimization and modeling problem 1 [calculus]?

Question

A smokestack deposits soot on the ground with a concentration inversely proportional to the square of the distance from the stack. With two smokestacks 20 miles apart, he concentration of the combined deposits on the line joining them, at a distance [x] from one stack is given by,

S = (k1/x^2) + (k2/[20-x]^2)

where k1 and k2 are positive constants which depend on the quantity of smoke each stack is emitting. If k1 = 7k2, find the point on the line joining the stacks where the concentration of the deposit is a MINIMUM.

help help help!

Answer 1

Take the derivative of S and set it equal to 0. Use the fact that k1 = 7*k2, and you'll find that the answer does NOT depend on the actual value of k1, because k1 just factors out. The minimum obviously doesn't occur at an endpoint, because S goes to infinity at the endpoints. So where else can the minimum be? :)