The minimum distance is the perpendicular distance. So establish the distance from the origin to the line using the distance formula. The distance here is: <span><span>d2</span>=(x−0<span>)^2</span>+(y−0<span>)^2 </span> =<span>x^2</span>+<span>y^2 </span></span> To minimize this function d^2 subject to the constraint, <span>2x+y−10=0 </span>If we substitute, the y-values the distance function can take will be related to the x-values by the line:<span>y=10−2x </span>You can substitute this in for y in the distance function and take the derivative: <span>d=sqrt [<span><span><span>x2</span>+(10−2x<span>)^2] </span></span></span></span> d′=1/2 (5x2−40x+100)^(−1/2) (10x−40)<span> </span>Setting the derivative to zero to find optimal x, <span><span>d′</span>=0→10x−40=0→x=4 </span> This will be the x-value on the line such that the distance between the origin and line will be EITHER a maximum or minimum (technically, it should be checked afterward). For x = 4, the corresponding y-value is found from the equation of the line (since we need the corresponding y-value on the line for this x-value).
Then y = 10 - 2(4) = 2.
So the point, P, is (4,2).
