1. ## Distance problems?

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

2. ## Re: Distance problems?

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?

3. ## Re: Distance problems?

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

4. ## Re: Distance problems?

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

5. ## Re: Distance problems?

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

6. ## Re: Distance problems?

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.