Respuesta :
Answer:
(a) [tex]f(7,0) = 49[/tex]
(b) The set of real numbers [tex]\mathbb{R}^2[/tex]
(c) The set of positive real numbers [tex]\mathbb{R}^+[/tex]
Step-by-step explanation:
To evaluate [tex]f(7,0)[/tex], simply substitute[tex]x=7[/tex] and [tex]y=0[/tex] into the given function [tex]f(x,y) = x^2 e^{2xy}[/tex]. This will give [tex]49[/tex].
The domain of [tex]f(x,y)[/tex]is the set of values of [tex]x[/tex] and [tex]y[/tex] for which [tex]f(x,y)[/tex] is defined. In this case, there is no value of [tex]x[/tex] or [tex]y[/tex] that makes the function [tex]f(x,y) = x^2e^{2xy}[/tex] undefined. Therefore, the domain is the set of all real numbers, that is [tex]x \in \mathbb{R}, y \in \mathbb{R}[/tex].
To determine the range of [tex]f(x,y)[/tex], let's use a logical approach.
First of all, irrespective of the value of [tex]x[/tex], the [tex]x^2[/tex] part of the function [tex]f(x,y)[/tex] is always positive.
For the exponential part of the function:
* if [tex]x[/tex] and [tex]y[/tex] are positive, [tex]e^{2xy}[/tex] is positive
* if one of [tex]x[/tex] or [tex]y[/tex] is negative, [tex]e^{2xy}[/tex] is positive
* if both [tex]x[/tex] and [tex]y[/tex] are negative, [tex]e^{2xy}[/tex] is positive.
Therefore, since the function is always positive irrespective of the values of [tex]x[/tex] and [tex]y[/tex], we can conclude that the range of [tex]f(x,y)[/tex] is the set of positive real numbers.
