the nearest point sort of problems

**saskadimova** · March 4th 2009, 05:10 PM

It is a minimization problem and i don't really understand them. Please give me some reasonable solution on the following one:

Find the point (x, y) belong toR2 on the graph of y = x^(1\2) nearest to the point (4, 0).

Thanks X 100000

**Soroban** · March 4th 2009, 08:57 PM

Hello, saskadimova!

Find the point $(x,y)$ on the graph of $y\,=\,x^{\frac{1}{2}}$ nearest to the point $(4, 0).$

Perhaps a sketch will help . . .

Code:

        |
        |             *
        | (x,y) *
        |   o
        | *  \
        |*    \d
        |      \
    - - * - - - o- - - - -
        |     (4,0)

The point has coordinates $(x,y) \,=\,\left(x,\sqrt{x}\right)$

We want its distance from $(4,0)$ to be a minimum.

The distance is: . $d \;=\;\sqrt{(x-4)^2 + \left(\sqrt{x} - 0\right)^2}$

. . So we have: . $d \;=\;\left(x^2-7x+16\right)^{\frac{1}{2}}$

Then: . $d\,' \;=\;\tfrac{1}{2}\left(x^2-7x+16\right)^{-\frac{1}{2}}(2x-7) \quad\Rightarrow\quad \frac{2x-7}{2\sqrt{x^2-7x+16}} \;=\;0$

Hence: . $2x-7\:=\:0 \quad\Rightarrow\quad x \:=\:\frac{7}{2}$

And: . $y \:=\:\sqrt{\frac{7}{2}} \:=\:\frac{\sqrt{14}}{2}$

Therefore, the nearest point is: . $\left(\frac{7}{2},\:\frac{\sqrt{14}}{2}\right)$

Thread: the nearest point sort of problems

LinkBack

Thread Tools

Display

the nearest point sort of problems

Similar Math Help Forum Discussions

Closest point problems

Nearest and farthest point

How to find the point on the curve nearest to A(3,0)?

solving critical point problems??

Help me to sort out this maths problems!!!!!

Search Tags