Answer:
Explanation:
1. Find the critical points:
To find the critical points, we need to find the points where the partial derivatives of the function are equal to zero.
The partial derivative with respect to x is ∂f/∂x = 2x + y.
The partial derivative with respect to y is ∂f/∂y = x + 2y + 7.
Setting both partial derivatives equal to zero, we have:
2x + y = 0 (equation 1)
x + 2y + 7 = 0 (equation 2)
2. Solve the system of equations:
Solving equations 1 and 2 simultaneously will give us the critical points.
From equation 1, we can solve for y in terms of x:
y = -2x.
Substituting this value into equation 2, we have:
x + 2(-2x) + 7 = 0
x - 4x + 7 = 0
-3x + 7 = 0
-3x = -7
x = 7/3.
Substituting the value of x back into y = -2x, we get:
y = -2(7/3)
y = -14/3.
So, the critical point is (7/3, -14/3).
3. Determine the nature of the critical point:
To determine if the critical point is a local maximum, local minimum, or saddle point, we need to evaluate the second-order partial derivatives of the function.
The second partial derivative with respect to x is ∂²f/∂x² = 2.
The second partial derivative with respect to y is ∂²f/∂y² = 2.
The mixed second partial derivative is ∂²f/∂x∂y = 1.
4. Apply the second derivative test:
Using the second partial derivatives, we can apply the second derivative test to determine the nature of the critical point.
Calculate the discriminant D = (∂²f/∂x²)(∂²f/∂y²) - (∂²f/∂x∂y)²:
D = (2)(2) - (1)² = 4 - 1 = 3.
Since the discriminant D is positive and (∂²f/∂x²) > 0, the critical point (7/3, -14/3) is a local minimum.
To graph the function, you can use three-dimensional graphing software. Set the domain and viewpoint to reveal all the important aspects of the function. The graph will show the shape of the function and the location of the local minimum.