# Thread: Point on the graph minimize problem

1. ## Point on the graph minimize problem

Consider the function below.
Find the point on the graph of function that is closest to the point (-3, 12).
( , 2)

I know you have to use the distance formula and plug in your equation in for the y value. Differentiate simplify and set it equal to zero to minimize and solve for x. I keep getting stuck on the differentiation and solving for x.

thanks in advance for any help.

2. Here's a very useful hint. The distance formula has a square root in it, as you know, and thus is frustrating to differentiate. However take note of the fact that if you minimize $\displaystyle D^2$, you also minimize D. So try that out and if you're still stuck show your work and we'll find your error.

3. Hello, thebacontalks2me!

Consider the function: .$\displaystyle f(x) \:=\:(x+9)^2 + 9$

Find the point on the graph of $\displaystyle f(x)$ that is closest to the point $\displaystyle A(\text{-}3, 12)$
Here's another useful procedure . . .

Let the point be $\displaystyle P(x,y)$, where $\displaystyle y \:=\:(x+9)^2 + 9 \:=\:x^2 + 18x + 90$

Using Jameson's hint, let: .$\displaystyle D \;=\;(x+3)^2 + (y - 12)^2$

. . Differentiate: .$\displaystyle D' \;=\;2(x+3) + 2(y-12)\!\cdot\!y' \;=\;0$

Since $\displaystyle y-12\:=\:x^2+18x+78$, and $\displaystyle y' \:=\:2x + 18$

. . we have: .$\displaystyle 2(x + 3) + 2(x^2+18x+78)(2x+18) \;=\;0$

. . which simplifies to: .$\displaystyle 2x^3 + 54x^2 + 481x + 1407 \:=\:0$

. . which factors: .$\displaystyle (x+7)(2x^2 + 40x + 201) \;=\;0$

. . and has the real root: .$\displaystyle x\:=\:\text{-}7$

Then: .$\displaystyle y \:=\:(\text{-}7)^2 + 18(\text{-}7) + 90 \:=\:13$

Therefore, the nearest point is: .$\displaystyle (\text{-}7,\,13)$