Answer with Step-by-step explanation:
We are given that a function
[tex]f(x,y)=-8x^2+6y^2-14[/tex]
We have to find the critical points and find the function has local maximum, local minimum, or saddle point using second derivative test.
Differentiate w.r.t x
[tex]f_x=-16x[/tex]
[tex]f_{xx}=-16[/tex]
Differentiate function w.r.t y
[tex]f_y=12y[/tex]
[tex]f_{yy}=12[/tex]
[tex]f_{xy}=0[/tex]
To find the critical point
Substitute [tex]f_x=0,f_y=0[/tex]
[tex]-16x=0\implies x=0[/tex]
[tex]12y=0\implies y=0[/tex]
The critical point is (0,0).
Value of D at critical point (a,b)
[tex]D=f_{xx}f_{yy}-(f_{xy})^2[/tex]
[tex]f_{xx}(0,0)=-16[/tex]
[tex]f_{yy}(0,0)=12,f_{xy}(0,0)=0[/tex]
Substitute the values then we get
[tex]D=(-16)(12)-0[/tex]
[tex]D=-192 <0[/tex]
[tex]D < 0, f_{xx}(0,0) < 0[/tex]
Therefore, the function has saddle point at (0,0) because when D < 0 the f(x,y) has saddle point at critical point [tex](x_0,y_0)[/tex].
