Find the point on the graph of the function $f ( x ) = \sqrt { x }$ that is closest to the point $( 18,0 )$
A) $\left( \frac { 35 } { 2 } , \sqrt { \frac { 35 } { 2 } } \right)$
B) $\left( \frac { 37 } { 2 } , \sqrt { \frac { 37 } { 2 } } \right)$
C) $\left( \frac { 35 } { 2 } , \sqrt { \frac { 37 } { 2 } } \right)$
D) $\left( \sqrt { \frac { 37 } { 2 } } , \frac { 35 } { 2 } \right)$
E) $\left( \sqrt { \frac { 35 } { 2 } } , \frac { 35 } { 2 } \right)$

Question

Find the point on the graph of the function $f ( x ) = \sqrt { x }$ that is closest to the point $( 18,0 )$ 
A) $\left( \frac { 35 } { 2 } , \sqrt { \frac { 35 } { 2 } } \right)$
B) $\left( \frac { 37 } { 2 } , \sqrt { \frac { 37 } { 2 } } \right)$
C) $\left( \frac { 35 } { 2 } , \sqrt { \frac { 37 } { 2 } } \right)$
D) $\left( \sqrt { \frac { 37 } { 2 } } , \frac { 35 } { 2 } \right)$
E) $\left( \sqrt { \frac { 35 } { 2 } } , \frac { 35 } { 2 } \right)$

Quizplus · Accepted Answer

The distance between the point $( x, f(x) )$ and the point $( 18,0 )$ is given by$$d = \sqrt{(x-18)^2 + f(x)^2} = \sqrt{(x-18)^2 + x}$$To find the point on the graph of $f(x) = \sqrt{x}$ that is closest to the point $( 18,0 )$, we need to minimize the distance function $d$.Taking the derivative of $d$ with respect to $x$, we get$$\frac{d}{dx} d = \frac{1}{2} \left( \frac{2(x-18)}{d} + \frac{1}{d} \right) = \frac{x-18 + 1}{d} = \frac{x-17}{d}$$Setting the derivative equal to zero, we get$$\frac{x-17}{d} = 0 \Rightarrow x = 17$$Substituting $x = 17$ into $f(x) = \sqrt{x}$, we get$$f(17) = \sqrt{17}$$Therefore, the point on the graph of $f(x) = \sqrt{x}$ that is closest to the point $( 18,0 )$ is $( 17, \sqrt{17} )$.

Find the Point on the Graph of the Function f(x)=xf ( x ) = \sqrt { x }f(x)=x​

Find the Point on the Graph of the Function $f ( x ) = \sqrt { x }$