Final answer:
The function f(x, y) = (x² y²)e⁻⁰x does not have local extrema at infinity. The local minimum occurs at the origin (0,0), and there are no local maxima at large values of x and y due to the exponential term.
Step-by-step explanation:
When analyzing the function f(x, y) = (x² y²)e⁻⁰x for local extrema, we first consider the critical points where the first derivatives of the function with respect to both variables are zero. However, for this function, the critical point at (0,0) does not provide enough information to conclusively determine whether it's a local maximum or minimum without further analysis using second-order partial derivatives.
But by observing the behavior of the exponential function and the x²y² term, we can infer that as x and y approach infinity, the function will decrease towards zero because the exponential term will dominate and rapidly decrease. Hence, the function does not have a local maximum or minimum at infinity. Instead, the local minimum occurs at the origin, which is consistent with the exponential decay away from the origin.
The function increases as x and y move away from the origin and then starts to decrease due to the exponential term. So, there are no local maxima at large values of x and y.