Question

Let Z = f(x,y) = 3x^3 - 5y^2 - 225x + 70y + 23. (i) Find the stationary points of z. (ii) Determine if at these points the function is at arelative maximum, relative minimum, infixion point, or saddle point.

Answer 1

Z=f(x,y)=3x²-5y²-225x+70y+23

$\tfrac{\partial~f(x,y)}{\partial~x} =9x^{2} -225 =0$

x_o =$\sqrt{\dfrac{225}{9}}$ =5

$\tfrac{\partial~f(x,y)}{\partial~y} =-10 y +70 =0$

y=-7

D=$\begin{vmatrix} \tfrac{\partial{^2}~f(x_o,y_o)}{\partial~x^{2}} & \tfrac{\partial{^2}~f(x_o,y_o)}{\partial~y{\partial~x}} \\ \tfrac{\partial{^2}~f(x_o,y_o)}{\partial~x~\partial~y} & \tfrac{\partial{^2}~f(x_o,y_o)}{\partial~y^{2}} \end{vmatrix}$

D=$\begin{vmatrix} 90 & 0 \\ 0 & -10 \end{vmatrix}$

D=90*-10=-900

D<0,hence the stationary point (x_0,y₀) =(5,-7) is a saddle point.