Questions: Find the absolute extrema of the function (f(x, y)=5 x^2-5 x y+4 y^2+x-2 y+4) on the domain defined by (-1 leq x leq 2) and (-3 leq y leq 2). Please show your answer to at least 4 decimal places. Absolute Maximum: Absolute Minimum:

Solution Steps

To find the absolute extrema of the function \( f(x, y) = 5x^2 - 5xy + 4y^2 + x - 2y + 4 \) on the given domain, we need to evaluate the function at the critical points within the domain and at the boundary points. The critical points are found by setting the partial derivatives of \( f \) with respect to \( x \) and \( y \) to zero. Then, we evaluate \( f \) at these points and at the boundary points to determine the absolute maximum and minimum values.

To find the critical points of the function \( f(x, y) = 5x^2 - 5xy + 4y^2 + x - 2y + 4 \), we set the partial derivatives \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \) to zero. The optimization process yields a critical point at approximately \( (0.0364, 0.2727) \) with a function value of \( f(0.0364, 0.2727) \approx 3.7455 \).

Next, we evaluate the function at the boundary points defined by the constraints \( -1 \leq x \leq 2 \) and \( -3 \leq y \leq 2 \). The boundary points are:

• \( (-1, -3) \) yielding \( f(-1, -3) = 35 \)
• \( (-1, 2) \) yielding \( f(-1, 2) = 30 \)
• \( (2, -3) \) yielding \( f(2, -3) = 98 \)
• \( (2, 2) \) yielding \( f(2, 2) = 18 \)

We compile the function values from the critical point and the boundary points:

• \( f(0.0364, 0.2727) \approx 3.7455 \)
• \( f(-1, -3) = 35 \)
• \( f(-1, 2) = 30 \)
• \( f(2, -3) = 98 \)
• \( f(2, 2) = 18 \)

The absolute maximum value is \( 98 \) and the absolute minimum value is \( 3.7455 \).

Final Answer