- #1
whkoh
- 29
- 0
The numbers x and y satisfy [itex]0 < x \leq a^2, 0 < y \leq a^2, xy \geq a^2[/itex] where [itex]a \geq 1[/itex].
By sketching suitable graphs or otherwise, show that
[tex]x + y \geq 2a[/tex] and [itex]x \leq a^{2}y \leq a^{4}x[/itex]
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I don't know what to sketch (tried [itex]x \leq 1, y \leq 1, xy \leq 1[/itex]), so I tried algebraic methods.
For the 1st:
[tex]y < x+y[/tex]
[tex]y^2 < x^2 + 2xy + y^2[/tex]
[tex]x^2 + 2xy > 0[/tex]
For the second one:
[tex]x \leq a^2 \leq xy[/tex]
[tex]a^2 \leq xy \leq a^{2}y[/tex]
I'm really lost on this.
By sketching suitable graphs or otherwise, show that
[tex]x + y \geq 2a[/tex] and [itex]x \leq a^{2}y \leq a^{4}x[/itex]
---
I don't know what to sketch (tried [itex]x \leq 1, y \leq 1, xy \leq 1[/itex]), so I tried algebraic methods.
For the 1st:
[tex]y < x+y[/tex]
[tex]y^2 < x^2 + 2xy + y^2[/tex]
[tex]x^2 + 2xy > 0[/tex]
For the second one:
[tex]x \leq a^2 \leq xy[/tex]
[tex]a^2 \leq xy \leq a^{2}y[/tex]
I'm really lost on this.
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