Week $5$

Consider the functions

$f(x,y,z)=x+3y+2z,$ and $,g(x,y,z)=x^2-y^2+z^2$

$(a)$ Write down the gradients gradf and gradg.

$(b)$ Use the method $of$ Lagrange multipliers $to$ find the minimum value $of$ the function $f(x,y,z)$

$on$ the implicitly defined surface $g(x,y,z)=-36.$

Consider the function $f(x,y)=x{y}^2-x^2-8y.$

$(a)$ Find the gradient vector gradf and the Hessian matrix $Hfoff.$

$(b)f$ has exactly one stationary point. Find $it,$ showing your working.

$(c)Is$ this stationary point a local minimum, a local maximum, $or$ a saddle point? Explain why.