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suppose that point A has coordinates (0,6), point B has coordinates (0,-6), and point C has coordinates (8,0). Determine the coordinates for the point P on the x-axis for which the sum of the distances from P to each of the three points A, B and C is as small as possible. I know i have to find when the derivitive of P = 0, but im having a hard time coming up with an equation. I drew a picture and found out the distance of AB to be 12, AC to be 10, and BC to be 10. P lies on some point between (O,O) and (0,8). im just not sure where to go from here and my teachers have barely spent much time on this and the textbook is useless like usual. Do i add all three distances together and go from there? thanks i really appreciate it.

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The distance from $P = (x,0)$ to $A$ will be the same as the distance from $P$ to $B$: $$\sqrt{36+x^2}$$

The distance from $P$ to $C$ will be $$\sqrt{(8-x)^2} = |8 - x| = 8-x$$

(Assuming that x is less than 8)

Can you take it from here?

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Leave the unknown point P's coordinates as ($x$,0). Write the distances AP, BP and CP as functions of $x$. Sum these three, then minimize this sum.

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