find all points where the function has any relative extrema. identify any saddle points.\nf(x,y)=2x²+3xy +…

find all points where the function has any relative extrema. identify any saddle points.\nf(x,y)=2x²+3xy + 4y² - 5x+2y\nfind all points where the function has relative maxima. select the correct choice below and fill in any answer boxes within your choice.\na. the point(s) is(are) □.\n(simplify your answer. type an ordered pair. use a comma to separate answers as needed.)\nb. there are no relative maxima.\nfind all points where the function has relative minima. select the correct choice below and fill in any answer boxes within your choice.\na. the point(s) is(are) □.\n(simplify your answer. type an ordered pair. use a comma to separate answers as needed.)\nb. there are no local relative.\nidentify any saddle points. select the correct choice below and fill in any answer boxes within your choice.\na. there is(are) saddle point(s) located at □.\n(simplify your answer. type an ordered pair. use a comma to separate answers as needed.)\nb. there are no saddle points.

find all points where the function has any relative extrema. identify any saddle points.\nf(x,y)=2x²+3xy + 4y² - 5x+2y\nfind all points where the function has relative maxima. select the correct choice below and fill in any answer boxes within your choice.\na. the point(s) is(are) □.\n(simplify your answer. type an ordered pair. use a comma to separate answers as needed.)\nb. there are no relative maxima.\nfind all points where the function has relative minima. select the correct choice below and fill in any answer boxes within your choice.\na. the point(s) is(are) □.\n(simplify your answer. type an ordered pair. use a comma to separate answers as needed.)\nb. there are no local relative.\nidentify any saddle points. select the correct choice below and fill in any answer boxes within your choice.\na. there is(are) saddle point(s) located at □.\n(simplify your answer. type an ordered pair. use a comma to separate answers as needed.)\nb. there are no saddle points.

Answer

Explanation:

Step1: Calculate the first - order partial derivatives

The function is (f(x,y)=2x^{2}+3xy + 4y^{2}-5x + 2y). The first - order partial derivative with respect to (x) is: (f_{x}=\frac{\partial f}{\partial x}=4x + 3y-5) The first - order partial derivative with respect to (y) is: (f_{y}=\frac{\partial f}{\partial y}=3x + 8y+2)

Step2: Find the critical points

Set (f_{x}=0) and (f_{y}=0), so we have the system of equations: (\begin{cases}4x + 3y-5=0\3x + 8y+2=0\end{cases}) From the first equation (4x+3y = 5), we get (x=\frac{5 - 3y}{4}). Substitute (x=\frac{5 - 3y}{4}) into the second equation (3\times\frac{5 - 3y}{4}+8y+2 = 0) (\frac{15-9y}{4}+8y+2 = 0) Multiply through by (4) to get (15-9y + 32y+8 = 0) (23y=-23), so (y=-1) Substitute (y = - 1) into (x=\frac{5 - 3y}{4}), we have (x=\frac{5-3\times(-1)}{4}=\frac{5 + 3}{4}=2) The critical point is ((2,-1))

Step3: Calculate the second - order partial derivatives

(f_{xx}=\frac{\partial^{2}f}{\partial x^{2}}=4), (f_{xy}=\frac{\partial^{2}f}{\partial x\partial y}=3), (f_{yy}=\frac{\partial^{2}f}{\partial y^{2}}=8)

Step4: Use the second - derivative test

The discriminant (D=f_{xx}f_{yy}-(f_{xy})^{2}) Substitute (f_{xx}=4), (f_{xy}=3), (f_{yy}=8) into the formula: (D=(4\times8)-3^{2}=32 - 9=23>0) And (f_{xx}=4>0)

Answer:

A. The point(s) is(are) ((2,-1)). B. There are no relative maxima. A. The point(s) is(are) ((2,-1)). B. There are no saddle points.