Find the maximum and minimum values of the function \( f(x, y)=2 x^{2}+3 y^{2}-4 x-5 \) on the domain \( x^{2}+y^{2} \leq 25 \). The maximum value of \( f(x, y) \) is: List the point(s) where the function attains its maximum as an ordered pair, such as \( (-6,3) \), or a list of ordered pairs if there is more than one point, such as \( (1,3),(-4,7) \). The minimum value of \( f(x, y) \) is: -7 List points where the function attains its minimum as an ordered pair, such as \( (-6,3) \), or a list of ordered pairs if there is more than one point, such as \( (1,3),(-4,7) \). \( (1,0) \)

To solve for the maximum and minimum values of the function \( f(x, y) = 2x^2 + 3y^2 - 4x - 5 \) within the constraint \( x^2 + y^2 \leq 25 \), we can analyze the function and boundary of the constraint. The critical points can be found using partial derivatives or by completing the square: - The function can be rewritten using completing the square for both \( x \) and \( y \). It shows that the function achieves its minimum of \(-7\) at the point \( (1, 0) \) inside the disk defined by the constraint. - To find the maximum, we need to evaluate the function on the boundary, \( x^2 + y^2 = 25 \). Substituting \( y^2 = 25 - x^2 \) into \( f \) allows us to find its extrema along the circle. In conclusion, the maximum value occurs at specific points on the boundary, typically found by evaluating \( f \) at various points such as \( (5,0), (-5,0), (0,5), (0,-5) \). The maximum value of \( f(x, y) \) is: 20 List the point(s) where the function attains its maximum: \( (5,0), (-5,0) \) The minimum value of \( f(x, y) \) is: -7 List points where the function attains its minimum as an ordered pair: \( (1,0) \)