Inequalities (Advanced)

Inequalities: core idea

• 1. Best constant: $\dfrac{a}{b}+\dfrac{b}{a}\ge k$ for positives: $k=$?

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• 2. Hölder / power mean: for $1,1$, QM-AM: $\sqrt{(1+1)/2}=$?

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• 3. Schur: for $a=b=c=1$, $a^3+b^3+c^3+3abc$ vs $a^2b+\cdots$: enter left−right at equality $0$.

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• 4. Min $x^2+4/x^2$ for $x>0$?

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• 5. Max $\sqrt{ab}$ for $a+b=6$?

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• 6. Tangent line method: $e^x\ge1+x$: at $x=0$ equality? $1/0$

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• 7. Cyclic $a/(b+c)$ Nesbitt ≥? For positives.

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• 8. Solve $x^2<16$: greatest integer $x$?

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• 9. uvw / symmetric: for $a+b+c=3$, $ab+bc+ca=3$, $abc=1$: are they all $1$? $1/0$

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