Happy new year guys! Don't ask me why I'm doing maths on New Years Eve

Question:

Find the stationary points and examine the following function for relative extrema and saddle points:

$\displaystyle f(x,y) = 4x^2 e^y - 2x^4 - e^{4y} $

$\displaystyle f_x = 8xe^y - 8x^3 = 0 $ which implies x=0 or $\displaystyle x=e^{\frac{y}{2}} $

$\displaystyle f_y = 4x^2e^y - 4e^{4y} = 0 $

substituting x=0 gives you end up getting $\displaystyle e^{4y} = 0 $ which is not defined. substituting $\displaystyle x=e^{\frac{y}{2}} $ gives you $\displaystyle 4e^{2y}(1-e^{2y}) = 0 $ which implies y=0.

So (1,0) is a stationary point - I am able to find the nature of this point, so don't worry about helping me with that.

From the $\displaystyle f_y = 0$ equation, you get $\displaystyle 4e^y (x^2 - e^{3y} ) = 0 $, so $\displaystyle x = e^{\frac{3y}{2}} $, substituting into the $\displaystyle f_x $ equation, you get $\displaystyle 8e^{\frac{5y}{2}} - 8e^{\frac{15}{2}}=0 $

$\displaystyle y= \frac{2}{5} = 3 $

So $\displaystyle ( e^{\frac{9}{2}}, 3) $ is another stationary point. Again I'm able to find the nature of this point.

These seem right because I've substituted them into the $\displaystyle f_x $ and $\displaystyle f_y $ equations and they work.

My concern is there must be more, because looking at the graph, there seem to be at least 2 maximum points and a minimum. Have I gone wrong somewhere or missed something? Thanks for taking the time to look at this.