Question #dbd28

1 Answer
Jan 14, 2016

Define the distance between the graph and the point as a function and find the minimum.

The point is #(3.5,1.871)#

Explanation:

To know how close they are, you need to know the distance. The Euclidean distance is:

#sqrt(Δx^2+Δy^2)#

where Δx and Δy are the differences between the 2 points. In order to be the nearest point, that point has to have the minimum distance. Therefore, we set:

#f(x)=sqrt((x-4)^2+(x^(1/2)-0)^2)#

#f(x)=sqrt(x^2-8x+16+(x^(1/2))^2)#

#f(x)=sqrt(x^2-8x+16+x^(1/2*2))#

#f(x)=sqrt(x^2-8x+16+x)#

#f(x)=sqrt(x^2-7x+16)#

We now need to find the minimum of this function:

#f'(x)=1/(2*sqrt(x^2-7x+16))*(x^2-7x+16)'#

#f'(x)=(2x-7)/(2*sqrt(x^2-7x+16))#

The denominator is always positive as a square root function. The numerator is positive when:

#2x-7>0#

#x>7/2#

#x>3.5#

So the function is positive when #x>3.5#. Similarly, it can be proved that it is negative when #x<3.5# Therefore, there function #f(x)# has a minimum at #x=3.5#, which means the distance is the least at #x=3.5# The y coordinate of #y=x^(1/2)# is:

#y=3.5^(1/2)=sqrt(3.5)=1.871#

Finally, the point where the least distance from (4,0) is observed is:

#(3.5,1.871)#