Question

Find the point on the curve \(y = \sqrt x \) that is closest to the point\(\)\(\left( {{\bf{3,0}}} \right)\).

Short answer

The points are\(\left( {\frac{5}{2},\;\sqrt {\frac{5}{2}} } \right)\).

Step-by-step solution

Given data

\(y = \sqrt x \)

Points \(\left( {3,0} \right)\)

 Step 2: Solution

Let\(\left( {x,y} \right)\)be the point on curve

Distance between\(\left( {x,y} \right)\)and\(\left( {3,0} \right)\)is:

Therefore, assume\(f\left( x \right) = {\left( {x - 3} \right)^2} + x\).

We have to minimize\(f\left( x \right)\).

Since\(f\left( x \right)\)is minimum, this implies that\(f\left( x \right) = 0\). Therefore,

\(\begin{aligned}{c}2\left( {x - 3} \right) + 1 &= 0\\\left( {x - 3} \right) &= - \frac{1}{2}\\x &= 3 - \frac{1}{2}\\x &= \frac{5}{2}\end{aligned}\)

Find y using\(y = \sqrt x \)as:

\(y = \sqrt {\frac{5}{2}} \)

Hence, \(\left( {\frac{5}{2},\;\sqrt {\frac{5}{2}} } \right)\)is closest to \(\left( {3,0} \right)\).