# Math Help - optimzation problem #1

1. ## optimzation problem #1

Determine the minimal distance from point (-3,3) to the curve given by y=(x-3)^2

2. Originally Posted by skeske1234
Determine the minimal distance from point (-3,3) to the curve given by y=(x-3)^2
points are $(-3,3)$ and $(x, (x-3)^2)$

use the distance formula

3. Let [mathA(-3,3), \ P(x,(x-3)^2)[/tex]

$AP^2=(x+3)^2+[(x-3)^2-3]^2=x^4-12x^3+49x^2-66x+45=f(x)$

$f'(x)=2(2x^3-18x^2+49x-33)=2(x-1)(2x^2-16x+33)$

$f'(x)=0\Rightarrow x=1$

If $x<1\Rightarrow f'(x)<0\Rightarrow$ f decreasing.

If $x>1\Rightarrow f'(x)>0\Rightarrow$ f increasing.

Then $x=1$is a point of minimum. The minimum of f is $f(1)=17$. The minimum distance is $\sqrt{17}$