Answer to Question #288854 in Calculus for Pankaj

Question

The range of the function f(x, y) =√81-9x²-9y² is [0, 8] . Say true or false.

Answer 1

$$f(x, y)=\sqrt{81-9x^2-9y^2}$$

$$x^2+y^2\geq0, x, y\in \R$$

$$-9x^2-9y^2\leq0$$

$$81-9x^2-9y^2\leq81$$

We need that $81-9x^2-9y^2\geq 0$

Then

$$0\leq\sqrt{81-9x^2-9y^2}\leq\sqrt{81}$$

$$0\leq\sqrt{81-9x^2-9y^2}\leq9$$

The range of the function $f(x, y)$ is $[0,9].$

Therefore the statement that the range of the function

$$f(x, y)=\sqrt{81-9x^2-9y^2}$$

is $[0, 8]$ is False.