Distance problems?

**camjenson** · December 9th 2012, 07:31 PM

y=x^(1/2). what's the shortest distance from that curve to the point (1/2,0)?

**MarkFL** · December 9th 2012, 08:08 PM

Let $\left(x,x^{\frac{1}{2}} \right)$ be an arbitrary point on the curve. It will be simpler to minimize the square of the distance between this arbitrary point and the given fixed point. Can you state the square of this distance?

**camjenson** · December 9th 2012, 08:15 PM

would this use the distance formula? or am i way off?

**MarkFL** · December 9th 2012, 08:17 PM

You are spot on! Since our objective function is a distance, then that's a great place to start!

**camjenson** · December 9th 2012, 08:45 PM

so how do i use that? what do i plug in?

**MarkFL** · December 9th 2012, 09:15 PM

We have two points:

$\left(x,x^{\frac{1}{2}} \right)$ and $\left(\frac{1}{2},0 \right)$ .

Now, the distance formula tells us the distance $d$ between the two points $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$ is:

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

So, choose either of our two points to be $P_1$ and the other as $P_2$ and plug into the equivalent:

$d^2=(x_2-x_1)^2+(y_2-y_1)^2$

Then simplify and optimize by differentiation. Be sure to demonstrate you have minimized by using an appropriate test.

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