minimization problem of 2 variables

Question

for what values of a and b is the following expression minimum

$$D= \sqrt{ (a-3)^2 + (b-5)^2 } + \sqrt{ (a-7)^2+ (b-9)^2 } + \sqrt{ (a-4)^2 + (b-1)^2 }$$

Please do let me know how to solve this. I need to find the point$(a,b)$ that is closest to the other points.

Answer 1

Probably not the sort of solution you are looking for, but this problem can be geometrically interpreted as follows: Let $\Delta ABC$ be the triangle with vertices at (3,5), (7,9), and (4,1). Find a point $(a,b)$ such that the sum of the distances from $(a,b)$ to the three vertices of $\Delta ABC$ is minimized.

That point is known as the Fermat point of the triangle. If you follow that link, you will find the following (crucial) fact:

When a triangle has an angle greater than 120°, the Fermat point is sited at the obtuse-angled vertex.

You happen to be dealing with a triangle that has exactly that setup.