Partial Derivatives

Question

Find the point(s) on the surface z²=xy+1 which are closest to the point (7, 11, 0).

List points as a comma-separated list, (e.g., (1,1,-1), (2, 0, -1), (2,0, 3))

Answer 1

Find the point(s) on the surface z²=xy+1 which are closest to the point (7, 11, 0).

The distance of any point on the surface from the point (7, 11, 0) is

D(x,y) = { [ x -7]² + [ y -11 ]² + [ z -0 ]² } ^1/2

D(x,y) = { [ x -7]² + [ y -11 ]² + z ² } ^1/2

D(x,y) = { [ x -7]² + [ y -11 ]² + z ² } ^1/2

D(x,y) = { [ x -7]² + [ y -11 ]² + x y +1 } ^1/2

To minimize D suffices to minimize

d (x ,y ) = [ x -7]² + [ y -11 ]² + x y +1

∂ d / ∂ x =2 x + y - 14

∂ d / d y = 2 y +x - 22

Solving the system 2 x + y - 14 = 0 and 2 y +x - 22 = 0 we get for x = 2 and for y = 10

The Hessian being 3 > 0 and d_xx and d_yy positive makes sure that the points

( 2 , 10, √21 ) and ( 2 , 10, -√21 ) are the points of interest