Question

Find the exact extreme values of the function z=f(x,y)=3x 2 −2xy+4y 2 −6x−20y+9 subject to the following constraints: 0≤x≤5 0≤y≤6 ​ Start by listing all nine candidates, including their z values, in the form (x,y,z) : First, list the four corner points and order your answers from smallest to largest x, then from smallest to largest y. Next find the critical point. 5) (x,x,x) Lastly, find the four boundary points and order your answers from smallest to largest x, then from smallest to largest y. 6) ([ ′ ,− ′ ,−) 7) ([□ ′ ,− ′ ) 8) ([L ′ , ′ ,−1) 9) ( ′ , Finally, find the extreme values: f min ​ =x at (x,y)=(x,x) f max ​ = at (x,y)=(3×,) ​

Answer 1

To find the extreme values of a function z=f(x,y) given constraints on x and y, evaluate the function at the corner points, find the critical point, and evaluate the function on the domain boundaries. Then, compare all z-values to identify the minimum and maximum values of the function.

Step-by-step explanation:

To find the extreme values of the function z=f(x,y)=3x^2−2xy+4y^2−6x−20y+9 subject to the constraints 0≤x≤5 and 0≤y≤6, we start by evaluating the function at the corner points of the domain.

(0,0,z) where z = f(0,0)

(5,0,z) where z = f(5,0)

(0,6,z) where z = f(0,6)

(5,6,z) where z = f(5,6)

Next, we find the critical point by setting the partial derivatives of f with respect to x and y to zero, and solving for x and y within the given constraints.

Finally, we evaluate the function on the boundaries of the domain by letting x or y be at their maximum or minimum values while the other variable ranges within its constraints. This gives us four more points:

(x',0,z) where x' is a critical value of x on the boundary 0≤x≤5

(x',6,z) where x' is a critical value of x on the boundary 0≤x≤5

(0,y',z) where y' is a critical value of y on the boundary 0≤y≤6

(5,y',z) where y' is a critical value of y on the boundary 0≤y≤6

After evaluating all nine points, we identify the minimum value (f min) and maximum value (f max) of the function within the given constraints by comparing the z-values.