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(Steps Shown) Show that the function f(x, y)=3 x e^y-x^3-e^3 y has only one critical point, at (1,0). Prove that it is a local maximum (yet the function has

Question: Show that the function \(f(x, y)=3 x e^{y}-x^{3}-e^{3 y}\) has only one critical point, at \((1,0)\). Prove that it is a local maximum (yet the function has no absolute maximum, as \(\left.\lim _{x \rightarrow-\infty} f(x, 0)=+\infty\right)\).

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