If we define to be the total illumination, then
where
Because approaches at and is differentiable everywhere else, the minimum will occur at a point at which .
The illumination at a point is inversely proportional to the square of the distance of the point from the light source and directly proportional to the intensity of the light source. If two light sources are 20 feet apart and their intensities are 40 and 30 respectively, at what point between them will the sum of their illuminations be a minimum?
Let x be the distance from the brighter source at which the sum of the illuminations is a minimum. Then x = how many feet?