prove $(a + x)^{1/2} + (a - x)^{1/2} \gt a$ for any real $a\gt 0$.
Algebraic Inequality
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6@Shawn: For all $x$? Surely not, put $x=a$. The assertion becomes $\sqrt{2a} >a$, which is false if $a \ge 2$. Can you reformulate the question? Maybe you are trying to find what values of $x$ satisfy the inequality? – 2011-05-05
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0Oops sorry the question is to solve for such for x to make this true. – 2011-05-05
2 Answers
Obviously $-a\lt x\lt a$ (Comment of Shawn).
Squaring both sides... $(a+x)+(a-x)+2\sqrt{a^2 - x^2} = 2a + 2\sqrt{a^2-x^2} > a^2$
or $2\sqrt{a^2-x^2} > a(a-2)$
(When $a\geq2$)
$4(a^2-x^2) > a^2 (a^2 - 4a + 4)$
or $a^4 - 4a^3 + 4x^2 < 0$
or $x^2 < a^3 - a^4/4$
or $-a \sqrt{a - a^2/4} < x < a \sqrt{a - a^2/4}$.
(When $a\leq2$) (Comment of Arturo Magidin)
Since $a(a-2)\lt0$, $-a\lt x \lt a$ is enough.
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0Ah thanks for the solution, I guess I forgot you could simplify roots by multiplying their inner values. – 2011-05-05
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1Let $a=135/128$ then your equation gives $x>-\frac{405 \sqrt{5655}}{32768}\approx -0.9294$ but $x=-1$ works... – 2011-05-05
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1The third step assumes that $a-2\gt 0$ when it passes from $2\sqrt{a^2-x^2}\gt a(a-2)$ to $4(a^2-x^2)\gt a^2)(a-2)^2$. – 2011-05-05
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0@user3123, @Arturo : oops.. – 2011-05-05
Squaring both sides, we obtain the equivalent inequality $$2\sqrt{a^2-x^2} >a^2-2a$$ If $a^2-2a<0$, that is, if $0, the inequality automatically holds for all $x$ at which the left-hand side is defined, that is, for all $x$ such that $|x| \le a$.
So now consider the case $a\ge 2$. Then the inequality $2\sqrt{a^2-x^2}>a^2-2a$ is equivalent to $4(a^2-x^2)>(a^2-2a)^2$. This simplifies to $$4x^2 <4a^3-a^4$$ or equivalently $|x|<(a/2)\sqrt{4a-a^2}$. Note that in particular there are no solutions if $a \ge 4$.
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1This one is the correct answer. – 2011-05-05