Question

The Partial Derivative Test

find the critical values for f(x,y)=6xy^2 -2x^3 - 3y^4 test and classify the nature of the extrema

Answer 1

Calculus is life lol you don't question you just do it XD JK.

Answer 2

what have you tried?

Answer 3

fx=6y^2-6x^2 fy=12xy-12y^3

Answer 4

i set fx equal to 0 and got x=y

Answer 5

can you first spot the "obvious" solution

Answer 6

no

Answer 7

notice x=0 and y=0 is a solution

Answer 8

then

\[f_y=12xy-12y^3 =0\]

\[y(12x-12y^2)=0\]

so

\[12x-12y^2=0\]

Answer 9

\[f_x=6y^2-6x^2=0 \]

so

\[12y^2-12x^2=0 \]

Answer 10

add
\[12x-12y^2=0\]

and
\[12y^2-12x^2=0\]


then solve for x

Answer 11

what do you mean add?

Answer 12

if we know x=0 and y=0 then (0,0)

Answer 13

shouldnt we use the determinant fxx*fyy-(fx)^2 to test its extrema?