91Ó°ÊÓ

Are the statements true or false? Give reasons for your answer. The function \(f(x, y)=x^{2}+y^{2}\) has a global maximum on the region \(x^{2}+y^{2}<1\).

Give an example of: A nonlinear function having no critical points

Are the statements true or false? Give reasons for your answer. If \(P\) and \(Q\) are two distinct points in 2-space, and \(f\) has a global maximum at \(P,\) then \(f\) cannot have a global maximum at \(Q\).

Give an example of: A function \(f(x, y)\) with a local maximum at (2,-3,4)

Are the statements true or false? Give reasons for your answer. The function \(f(x, y)=\sin \left(1+e^{x y}\right)\) must have a global minimum in the square region \(0 \leq x \leq 1,0 \leq y \leq 1\).

Are the statements true or false? Give reasons for your answer. If \(P_{0}\) is a global minimum of \(f\) on a closed and bounded region, then \(P_{0}\) need not be a critical point of \(f\).

Give an example of: A function \(f(x, y)\) that has a maximum but no minimum on the constraint \(x+y=4\)

Are the statements true or false? Give reasons for your answer. If \(P_{0}\) is a local maximum or local minimum of \(f,\) and not on the boundary of the domain of \(f,\) then \(P_{0}\) is a critical point of \(f\).

Are the statements true or false? Give reasons for your answer. If \(f(x, y)\) has a local maximum at \((a, b)\) subject to the constraint \(g(x, y)=c,\) then \(g(a, b)=c\)

Are the statements true or false? Give reasons for your answer. The function \(f(x, y)=x^{2}-y^{2}\) has a local minimum at the origin.