Find Critical Points of f(x,y)?

critical points are when df(x,y)/dx=0 and df(x,y)/dy=0

(the actually del not d, but you get the point, differentiate with respect to x and y)

df(x,y)/dx = fx(x,y) = y-16xy-7y^2 = 0

df(x,y)/dy = fy(x,y) = x-8x^2-14xy = 0

solve the system of equations (i assume you can do this using algebra)

x1 = 0

y1 = 0

x2 = 0

y2 = 1/7

x3 = 1/24

y3 = 1/21

x4 = 1/8

y4 = 0

these are you critical points

Now we find the function D(x,y) = fxx(x,y)*fyy(x,y)-fxy(x,y)^2 to detemine type

fxx(x,y) = dfx(x,y)/dx = -16y

fyy(x,y) = dfy(x,y)/dy = -14x

fxy(x,y) = dfx(x,y)/dy = 1-16x-14y

D(x,y) = 256xy+224y^2-16y

D(0,0) = 0

D(0,1/7) = 16/7 > 0

D(1/24,1/21) = 16/63 > 0

D(1/8,0) = 0

plug in the points that can be determined to fxx(x,y)

fxx(0,1/7) = -16/7 < 0

fxx(1/24,1/21) = -16/21 < 0

the points (0,1/7) and (1/24,1/21) are maximums because fxx < 0 and D > 0, because D=0 for the other two points the type of critical point cannot be determined with this method.

edit: messed up the derivatives the first time. Check with a calculator this time