<s>[INST] How to find critical numbers in awkward function I have problem to find critical numbers in this awkward function with Euler's constants. 
$$f(x, y) =  e^{2 x+3 y} \left(8 x^2-6 x y+3 y^2\right)$$
Task: Find critical numbers and determine if it's maximum or minimum
 [/INST] We find $f_x$ and $f_y$ set them equal to zero and find the critical points.
$$f_x = 2 e^{2 x+3 y} \left(8 x^2-6 x y+3 y^2\right)+e^{2 x+3 y} (16 x-6 y)=0$$
$$f_y = 3 e^{2 x+3 y} \left(8 x^2-6 x y+3 y^2\right)+e^{2 x+3 y} (6 y-6 x)=0$$
Clearly the exponential terms do not give zeros (divide those out), so we are left with solving:
$$2 \left(8 x^2-6 x y+3 y^2\right)+ (16 x-6 y)=0$$
$$3 \left(8 x^2-6 x y+3 y^2\right)+ (6 y-6 x)=0$$
Solving these leads to two critical points:
$$(x, y) = (0, 0), \left(-\dfrac{1}{4}, -\dfrac{1}{2}\right)$$
You can now try to classify them.
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