Find the critical points of the given function and then determine whether it is a local maximum, local minimum, or saddle point. f(x, y) = 8x2 + 2xy + 5x + y2 + y + 6 critical point classification (x, y ) =

Therefore the critical point is (x,y)= .

Therefore is local minimum.

Step-by-step explanation:

Given function is

The partial derivatives of z are

and  

To find the critical point, setting the partial derivatives equal to zero.

∴16x+2y+5 =0......(1) and 2x+2y+1=0.......(2)

Now solving the above equation,

Subtract equation (2) from equation (1)

16x+2y+5-( 2x+2y+1)=0

⇒16x+2y+5-2x-2y-1=0

⇒8x+4=0

⇒ 8x = -4

Now putting the value of x in (2)

⇒2y=0

⇒y=0

Therefore the critical point is (x,y)= .

Second order partial derivatives are

, and

The discriminant

  =14>0

D = 14 >0, Then at is either local minimum or local maximum.

Since  , So the function is concave up.

Therefore is local minimum.