Question

Locate a point C on the x-axis such that AC+BC is minimized. C=(_,0). The endpoints of AB are A(0,2) and B(9,4)

Answer 1

Let point C be (x, 0), then
AC = sqrt((x - 0)^2 + (0 - 2)^2) = sqrt(x^2 + 4) and
BC = sqrt((x - 9)^2 + (0 - 4)^2) = sqrt(x^2 - 18x + 81 + 16) = sqrt(x^2 - 18x + 97)
AC + BC = sqrt(x^2 + 4) + sqrt(x^2 - 18x + 97)

For minimum AC + BC, d(AC + BC)/dx = 0
d(AC + BC)/dx = x/sqrt(x^2 + 4) + (2x - 18)/sqrt(x^2 - 18x + 97) = 0
x(x^2 - 18x + 97) = -(2x - 18)(x^2 + 4)
x^3 - 18x^2 + 97x = -(2x^3 + 8x - 18x^2 - 72) = -2x^3 + 18x^2 - 8x + 72
3x^3 - 36x^2 + 105x - 72 = 0
x^3 - 12x^2 + 35x - 24 = 0
x = 8, 3, 1

Therefore, point C = (8, 0) or (3, 0) or (1, 0)