Posted by Mishaka on .
Find the point on the graph of y = x^2 + 1 that’s closest to the point 8, 1.5. Hint: Remember the distance formula.

Calculus 
drwls,
Distance to point, squared, is:
R^2 = (x8)^2 + (y1.5)^2
= (x8)^2 + (x^2 0.5)^2
Solve for the x value when d(R^2)/dx = 0
d/dx [x^2 16x +64 + x^4 x^2 + 1/4] = 0
2x 16 +3x^3 2x = 0
3x^3 = 16
x = 1.747
y = 4.053 
Calculus  typo 
Steve,
Note that d/dx x^4 = 4x^3 not 3x^3
From there on, we have
2x  16 + 4x^3  2x = 0
4x^3 = 16
x = cbrt(4)
y = cbrt(16)+1
Or, looking at things in another way, the normal line from point (p,q) on the curve will have the shortest distance to (8,1.5)
At any point (p,q) on the curve, the slope is 2p, so the normal line has slope 1/2p
Now we have a point and a slope:
(yq)/(xp) = 1/2p
y  p^2  1 = (px)/2p
1.5 = (p8)/2p + p^2 + 1
3p = p  8 + 2p^3 + 2p
2p^3 = 8
p^3 = 4
p = cbrt(4)
q = cbrt(16)+1 
Thanks 
drwls,
Thanks Steve for noticing my error