Choose any point (k,k^2) with k>0 on the parabola y=x^2. Draw the normal line to the parabola at that point. Then there is a closed region defined by the parabola and the line. Find the value of k so the area of the region is minimized.
Note: A normal line is a line perpendicular to a tangent at the point of tangency.
(In reply to solution
and this is minimized when
4k^6+ k^4 -(1/4)k^2-(1/16)=0 *16
64k^6+ k^4 -4k^2-1=0
while multiplying by 16 you have left one member (k^4) unchanged
therefore the rest is wrong , although your solution followed the right track.
Please correct and explain way of solving - I suggest substitutingb m=k^2 to get a cubic equation.