Let M be a positive real number and let u, v: $\R^2\rightarrow\R$ be continuous functions satisfying $\sqrt{u(x,y)^2+v(x,y)^2}\ge M\sqrt{x^2+y^2}\ for\ all\ (x,y)\isin\R^2.$ Let F: $\R^2\rightarrow\R^2$ be given by F(x, y) = (u(x, y), v(x, y)) for (x, y) $\isin\R^2$ . Then which of the following is/are true?

Question

Let M be a positive real number and let u, v: $\R^2\rightarrow\R$  be continuous functions satisfying  $\sqrt{u(x,y)^2+v(x,y)^2}\ge M\sqrt{x^2+y^2}\ for\ all\ (x,y)\isin\R^2.$ Let F:  $\R^2\rightarrow\R^2$  be given by F(x, y) = (u(x, y), v(x, y)) for (x, y) $\isin\R^2$ . Then which of the following is/are true?

cdquestions Admin · Accepted Answer

The correct option is (C): If K is closed in $\R^2$ , then F(K) is closed in  $\R^2$ . and (D): If E is closed and bounded in  $\R^2$ , then F -1 (E) is closed and bounded in  $\R^2$ .

Answer

F is injective.

Answer

If K is open in $\R^2$, then F(K) is open in $\R^2$.

Answer

If K is closed in $\R^2$, then F(K) is closed in $\R^2$.

Answer

If E is closed and bounded in $\R^2$, then F^-1(E) is closed and bounded in $\R^2$.

The Correct Option is C, D

Solution and Explanation

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