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if :

$$-10

then : What is the range $xy$?

my try :

$$|x|<10\\|y|<70\\x^2<(10^2)\\y^2<(70)^2\\x^2+y^2<(10)^2+(70)^2\\(x+y)^2-2xy<(10)^2+(70)^2\\ \frac{(40)^2-(10)^2-(70)^2}{2}

is it right ?

1 Answers 1

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To have $xy$ maximal with a given sum you want $x=y$. Here we can't have that because of the restriction on $x$, so the greatest product we can get is $x=10,y=30,xy=300$. The most negative product we can get is $x=-10,y=50, xy=-500$, so $$-500 \le xy \le 300$$ You can plot a graph of $xy=x(40-x)$ over the range allowed. enter image description here

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    There is also $x=10, y=-50$ for the lower bound, and note that the inequalities in the question are strict. – 2017-01-25