Question

By examining the Hessian matrix, show that if f(x, y, z) has a local minimum at (x0, y0, z0),

then g(x, y, z) = −f(x, y, z) must have a local maximum at that point. Likewise, show that if f has

a local maximum, then g must have a local minimum at that point.

Answer 1

f has local minimum at(,Yo.20) eigen values of H, (*0,,2) eigen values of Hy (0..20) are neagtive eigen values of H, (..20) g has ocal maximum at (x,.Vo.z0) are positive are neagtive f has local maximum at (0.Vo,20) eigen values of H, (0.^o.20) are neagtive eigen values of H; (,,20) are positive eigen values of H, ( ) are positive g has ocal minimum at (x0,Vo,20