Question

How close does the curve y=sqrt{x} come to the point (3/2,0)?

Answer 1

What do you mean by "close"? Do you mean the minimum distance? Or are you asking for the distance from any point on the curve to \(\left(\dfrac{3}{2},0\right)\) ?

Answer 2

I am Asking... what is the shortest distance from a point on the curve y=sqrt{x} to the point (3/2,0)?

Answer 3

Let \((a,b)\) be a point on the curve \(y=\sqrt x\). This would mean \(b=\sqrt a\).

The distance between this point and \(\left(\dfrac{3}{2},0\right)\) is
\[\begin{align*}d&=\sqrt{\left(a-\frac{3}{2}\right)^2+\left(\sqrt a-0\right)^2}\\\\
&=\sqrt{a^2-3a+\frac{9}{4}+a^2}\\\\
&=\sqrt{2a^2-3a+\frac{9}{4}}\end{align*}\]
You want to find \(a\) that minimizes this \(d\). Take the derivative.
\[\begin{align*}\frac{dd}{da}&=\frac{4a-3}{2\sqrt{\left(a-\dfrac{3}{2}\right)^2+\left(\sqrt a-0\right)^2}}\end{align*}\]
Find the critical value \(a\).

Notice that whatever value you find that optimizes \(d\), it will also optimize \(d^*\) defined by
\[d^*=2a^2-3a+\frac{9}{4}\]
The takeaway here is that if you're given a complicated function of the form \(d=\sqrt{\cdots}\), then you can find the same critical value by considering the simpler function \(d^*=\cdots\).

Answer 4

The least distance is going to be a perpendicular to the tangent to the curve passing through the given point. Good enough?