Define f : $\R^2 → \R$ by f(x, y) = x 2 + 2y 2 - x for (x, y) ∈ $\R^2$ . Let D = {(x, y) ∈ $\R^2$ ∶ x 2 + y 2 ≤ 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 _________.

Question

Define f : $\R^2 → \R$  by f(x, y) = x 2 + 2y 2 - x for (x, y) ∈  $\R^2$ . Let D = {(x, y) ∈  $\R^2$  ∶ x 2 + y 2 ≤ 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 _________.

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The correct answer is : 5.

Correct Answer: 5

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