Question 859045
{{{drawing(300,300,-2,8,-5,5,grid(1),circle(4,0,0.2),circle(3,sqrt(3),0.2),line(4,0,3,sqrt(3)),graph(300,300,-2,8,-5,5,sqrt(x)))}}}
Use the distance formula to calculate the distance from (4,0) to any arbitrary point on the curve (x,y).
{{{D^2=(x-4)^2+(y-0)^2}}}
{{{D^2=(x-4)^2+y^2}}}
From the function,
{{{y=sqrt(x)}}}
{{{y^2=x}}}
Substitute,
{{{D^2=(x-4)^2+x}}}
{{{D^2=(x^2-8x+16)+x}}}
{{{L=D^2=x^2-7x+16}}}
To minimize the distance,you can also minimize the distance squared. So take the derivative of the distance squared with respect to x and set it to zero.
{{{dL/dx=2x-7}}}
{{{2x-7=0}}}
{{{2x=7}}}
{{{x=7/2}}}