f_X=4(X-2)^3

f_XX=12(X-2)^2

f_XY=0

f_Y=4(Y-3)^3

f_YY=12(Y-3)^2

F_YX=0

but how i will get the extreme value

Ok now set your 1st partial derivatives to 0

\(\displaystyle 4(x-2)^3 = 0\)

x = 2

\(\displaystyle 4(y-3)^3 = 0\)

y = 3

so your critical value is (2,3)

Now use

\(\displaystyle (f_{xx})(f_{yy}) -(f{xy})^2 = D\)

if D = 0 inconclusive

if D > 0 max or min

if D < 0 saddle point