Question

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

Answer 1

Answer with Step-by-step explanation:

We are given that a function

$f(x,y)=-8x^2+6y^2-14$

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

$f_x=-16x$

$f_{xx}=-16$

Differentiate function w.r.t y

$f_y=12y$

$f_{yy}=12$

$f_{xy}=0$

To find the critical point  

Substitute $f_x=0,f_y=0$

$-16x=0\implies x=0$

$12y=0\implies y=0$

The critical point is (0,0).

Value of  D at critical point (a,b)

$D=f_{xx}f_{yy}-(f_{xy})^2$

$f_{xx}(0,0)=-16$

$f_{yy}(0,0)=12,f_{xy}(0,0)=0$

Substitute the values then we get

$D=(-16)(12)-0$

$D=-192$

$D < 0, f_{xx}(0,0) < 0$

Therefore, the function has saddle point  at (0,0) because when D < 0 the f(x,y) has saddle point at critical point $(x_0,y_0)$.