Find the critical points of the following function. Use the Second Derivative Test to determine​ (if possible) whether each critical point corresponds to a local​ maximum, local​ minimum, or saddle point. If the Second Derivative Test is​ inconclusive, determine the behavior of the function at the critical points. f(x,y)=−8x2+6y2−14

Accepted Solution

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].