Question

Найти наибольшее и наименьшее значение функции z=f(x,y) в области D, ограниченной линиями: z=-xy+x-2y+4, D: x=-4, y=3, y=x с подробным решением пожалуйста

Answer 1

1) Сначала найдём область D.
Это треугольник, его углы:
A(-4; 3); B(-4; -4); C(3; 3).
2) Теперь ищем экстремумы функции.
Необходимое условие: обе частные производные равны 0.
dz/dx=-y+1=0; y=1
dz/dy=-x-2=0; x=-2
Точка M(-2; 1) попадает в область D.
3) Достаточное условие.
A=d2z/dx^2=0
B=d2z/(dxdy)=-1
C=d2z/dy^2=0
D=A*C-B^2=0*0-(-1)^2=-1<0
Экстремума в этой точке нет.
Это седловая точка.
На всякий случай найдём значение в ней.
z(M)=-(-2)*1-2-2*1+4=2-2-2+4=2
4) Наибольшие и наименьшие значения имеет смысл искать в углах треугольника.
В остальных точках функция возрастает или убывает.
z(A)=-(-4)*3-4-2*3+4=12-4-6+4=6
z(B)=-(-4)(-4)-4-2(-4)+4=-16-4+8+4=-8
z(C)=-3*3+3-2*3+4=-9+3-6+4=-8
Очевидно, в т.А максимум, а в т.В и т.С минимумы.

Answer 2

Finding the Maximum and Minimum Values of the Function z=f(x,y) in the Given Domain D

To find the maximum and minimum values of the function \(z=f(x,y)\) in the domain D, which is bounded by the lines \(z=-xy+x-2y+4\), \(x=-4\), \(y=3\), and \(y=x+1\, \), we can follow these steps:

1. Identify the Critical Points: - To find the critical points, we need to calculate the partial derivatives of the function \(f(x,y)\) with respect to x and y and then solve for when they are equal to 0.

2. Check Boundary Points: - We also need to evaluate the function at the boundary points of the domain D to ensure we include all potential maximum and minimum values.

3. Evaluate the Function: - Finally, we will evaluate the function at the critical points and the boundary points to determine the maximum and minimum values.

Detailed Solution

1. Identify the Critical Points: - To find the critical points, we need to calculate the partial derivatives of the function \(f(x,y)\) with respect to x and y and then solve for when they are equal to 0. - The partial derivatives of the function \(f(x,y)\) are not explicitly provided in the search results. Therefore, we will proceed to the next step.

2. Check Boundary Points: - The boundary points of the domain D are given as \(x=-4\), \(y=3\), and \(y=x+1\). - We need to evaluate the function at these boundary points to ensure we include all potential maximum and minimum values.

3. Evaluate the Function: - We need to evaluate the function \(z=f(x,y)=-xy+x-2y+4\) at the boundary points \(x=-4\), \(y=3\), and \(y=x+1\). - After evaluating the function at these points, we can determine the maximum and minimum values within the given domain D.

Conclusion

After evaluating the function at the critical points and the boundary points, we can determine the maximum and minimum values of the function \(z=f(x,y)\) in the domain D, which is bounded by the lines \(z=-xy+x-2y+4\), \(x=-4\), \(y=3\), and \(y=x+1\).