Find the minimum and maximum of the function $f(x, y) = (x + y)e^{xy}$ for all $-2 \leq x + y \leq 1$

Question

Let we have $$f(x, y) = (x + y)e^{xy}$$.

We want to find minimum and maximum values of $f$ for this set of values: $$\{\ (x, y)\ : -2 \leq x + y \leq 1\ \}$$

It seems like this task can be solved with looking at the first and second derivatives if we lock our values on some compact.

But here we have stripe goin for an infinity ($$x \to -\infty, y \to \infty$$ or $$x \to \infty, y \to -\infty$$ cases) and I couldn't parse the values ​​of the function at infinity.

Is there any ideas how to find minimum and maximum in this case? Ideas and hints would be really helpful for me.

Answer 1

We haven't any critical point inside the domain therefore we need to look at the boundary and we have

• for $x+y=1$

$$(x + y)e^{xy}=e^{x-x^2}$$

which has maximum at $x=\frac12$ that is $f\left(\frac12,\frac12\right)=\sqrt[4]e$

• for $x+y=-2$

$$(x + y)e^{xy}=-2e^{-2x-x^2}$$

which has minimum at $x=-1$ that is $f\left(-1,-1\right)=-2e$