Prove the following inequality
a^2/ (1 + a^4) ≤ 1/2 for any real a.
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Since x>0x>0 you can multiply through by xx to clear fractions without changing the sense of the inequality. This gives
x2+1≥2x
x2+1≥2x
Subtract 2x2x from each side:
x2−2x+1≥0
x2−2x+1≥0
or
(x−1)2≥0
(x−1)2≥0
Which is true, with equality only if x=1x=1 since squares are non-negative.
Now note that each of these steps can be reversed to take us from the last statement, which we know to be true, to the statement in your question, which you want to prove. Since x>0x>0 you can divide by xx. Write it out carefully and you will have your proof.
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Mar 9 '14 at 8:41
Mark Bennet
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Mar 9 '14 at 8:48
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Very nice solution, it highlights the conceptual part of multiplying an inequality by a positive factor and also teaches to eliminate the error produced by not changing the inequality sign when multiplied by a negative number. – Hawk Mar 9 '14 at 8:45
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x+1x−2=(x−−√−1x−−√)2
x+1x−2=(x−1x)2
x+1x−2⩾0
x+1x−2⩾0
Q.E.D. Note that this is merely a different way to write the well known A.M⩾G.MA.M⩾G.M
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Mar 9 '14 at 8:29
Guy
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Consider the function
f(x)=x+1x
f(x)=x+1x
Its first derivative
f′(x)=1−1x2
f′(x)=1−1x2
cancels for x=1x=1 and f(1)=2f(1)=2 is then an extremum. Its second derivative is
f′′(x)=2x3
f″(x)=2x3
is positive. Then x=1x=1 corresponds to a minimum and the inequality is always satisfied.
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Mar 9 '14 at 8:25
Claude Leibovici
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Feb 10 '15 at 14:30
Martin Sleziak
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It is the best answer. – sidneimv Mar 20 '14 at 20:58
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@sidneimv. Thank you ! – Claude Leibovici Mar 21 '14 at 5:15
It is the most comprehensive answer. It works for any function. – sidneimv Mar 21 '14 at 17:23
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HINT : All real squares are non-negative:
(x−−√−1x−−√)2≥0
(x−1x)2≥0
x−2+1x≥0
x−2+1x≥0
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Nov 12 '15 at 10:47
Yiyuan Lee
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By the inequality of arithmetic and geometric means,
x+y2≥xy−−√
x+y2≥xy
for two non-negative numbers xx and yy. Set y=1/xy=1/x to obtain the result.
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Nov 12 '15 at 10:49
Cm7F7Bb
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Other answers already explain the steps involved to find a proof for the stated inequality. However, none of them address the next step of proving, which is how you write down the proof.
−y) =2=2.
The main takeaway from this proof, to me, isn't the result itself, but the inequality at the heart of it: what might be called the "Geometric Inequality" 11+y≥1−y11+y≥1−y for y≥0y≥0 (and similarly, 11−y≥1+y11−y≥1+y as long as y<1y<1) has broad applicability for inequalities and estimations.
Answer:
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