Given:
f(x) = √(x^2-10x+314) + √(x^2+20x+325)
Determine the minimum value of |f(x)| for a real number x.
*** Adapted from a problem appearing in 2017 Singapore M.O. open.
Consider three points (5,17), (x,0), and (-10,-15).
Then sqrt[x^2-10x+314] is the distance from (5,17) to (x,0)
And sqrt[x^2+20x+325] is the distance from (x,0) to (-10,-15)
f(x) is the sum of the distances of these two segments.
(x,0) is a point on the x-axis, which goes between the two points (5,17) and (-10,-15).
So then the minimum sum is the straight line distance from (5,17) to (-10,-15).
That distance is sqrt[(5-(-10))^2+(17-(-15))^2)] = sqrt[1249] = 35.341, which is then the minimum value of f(x).
Geometric solution
