Çözüldü Coordinate Geometry - Differentiation

With two points A(1, 5), B(2, 4), and another one C on the line that bisects the first and third quadrants, what is the abscissa of C so that the sum of |AC| and |BC| would be minimum?
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Solution - 1:
C(x, x) must be on the line passing through A and B, so the equation of this line is (y - 5) / (5 - 4) = (x - 1) / (1 - 2), substituting x for y, and simplifying;
x - 5 = -x + 1 ⇒ x = 3
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Solution - 2
f(x) = |AC| + |BC| = [ (x - 1)^2 + (x - 5)^2 ]^0,5 + [ (x - 2)^2 + (x - 4)^2 ]^0,5
f'(x) = [ 2(x - 1) + 2(x - 5) ] / { [ (x - 1)^2 + (x - 5)^2 ]^0,5 } + [ 2(x - 2) + 2(x - 4) ] / { [ (x - 2)^2 + (x - 4)^2 ]^0,5 } = 0
(2x - 6) / [ (√2)(√x^2 - 6x + 13) ] + (2x - 6) / [ (√2)(√x^2 - 6x + 10) ] = 0
(x - 3)·[ √(x^2 - 6x + 10) + √(x^2 - 6x + 13) ] = 0
From the first factor, x = 3 is found, and there is no need to check for the real roots of the second factor that has none because the discriminant of either quadratic equation is less than zero.