Question

Define f : \(\R^2 → \R\) by
f(x, y) = x² + 2y² - x for (x, y) ∈ \(\R^2\).
Let D = {(x, y) ∈ \(\R^2\) ∶ x² + y² ≤ 1} and \(E=\left\{(x,y)\in \R^2:\frac{x^2}{4}+\frac{y^2}{9} \le 1\right\}.\)
Consider the sets
D_max = {(a, b) ∈ D ∶ f has absolute maximum on D at (a, b)}, 
D_min = {(a, b) ∈ D ∶ f has absolute minimum on D at (a, b)}, 
E_max = {(c, d) ∈ E ∶ f has absolute maximum on E at (c, d)}, 
E_min = {(c, d) ∈ E ∶ f has absolute minimum on E at (c, d)}.
Then the total number of elements in the set
D_max ∪ D_min ∪ E_max ∪ E_min
is equal to _________.

Answer 1

Correct Answer: 5

To solve this problem, we need to focus on finding the critical points of the function \(f(x, y) = x^2 + 2y^2 - x\) on each set \(D\) and \(E\), and then verify whether these points yield absolute maximum or minimum values.

1. Function Analysis:
We begin by finding the critical points of \(f\). Compute the partial derivatives:
\(\frac{\partial f}{\partial x} = 2x - 1\) and \(\frac{\partial f}{\partial y} = 4y\).
Setting each to zero gives critical points:

• \(2x - 1 = 0 \Rightarrow x = \frac{1}{2}\)
• \(4y = 0 \Rightarrow y = 0\)

The critical point is \((\frac{1}{2}, 0)\).

2. Analyze on Set \(D\):
\(D = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 \leq 1\}\) is a disk of radius 1 centered at origin. Evaluate \(f\) at the boundary points:

• Consider the boundary \(x^2 + y^2 = 1\), use Lagrange Multipliers or substitution to evaluate.

\(f(x, y) = x^2 + 2y^2 - x\) subject to \(x^2 + y^2 = 1\).
Simultaneously solve:

• \((2x - 1) + 2\lambda x = 0\)
• \(4y + 2\lambda y = 0\)
• \(x^2 + y^2 = 1\)

Check points like \((1, 0)\) and \((-1, 0)\):

• \(f(1, 0) = 1^2 + 2(0)^2 - 1 = 0\)
• \(f(-1, 0) = (-1)^2 + 2(0)^2 + 1 = 2\)

3. Analyze on Set \(E\):
\(E = \left\{(x, y) \in \mathbb{R}^2: \frac{x^2}{4} + \frac{y^2}{9} \leq 1\right\}\) is an ellipse.

• Similar Lagrange Multipliers approach for ellipse yields boundary critical points.

Check axes intersections and major points inside ellipse:

• Boundary at \((4, 0)\) and \((-4, 0)\) gives \(f(4, 0) = 16 - 4 = 12\).
• Boundary at \((0, 3)\) and \((0, -3)\) gives \(f(0, 3) = 18\).

4. Compilation:
Understanding the behavior of \(f\) on both sets shows critical points.

5. Conclusion:
Calculate number of maxima/minima from boundary and critical points. From above, explicit check:
\((\frac{1}{2}, 0) \Rightarrow \text{min of } f\) and \((1, 0)\), \((-1, 0)\), boundary extreme \((-4, 0)\), \(f(-4, 0) = 18\).
The total number of distinct critical (maxima/minima) points is 5.

Therefore, the total number of elements in the set \(D_{max} \cup D_{min} \cup E_{max} \cup E_{min}\) is 5.