examday

/TMUA

Use some or all of the lines below to prove the inequality

holds if $a>0$ and $b>0$.

• 1. $(a+b)\left(\dfrac1a+\dfrac4b\right)\ge9\iff\dfrac ba+\dfrac{4a}b+5\ge9$
• 2. $\dfrac ba+\dfrac{4a}b+5\ge9\iff\dfrac ba+\dfrac{4a}b-4\ge0$
• 3. $\dfrac ba+\dfrac{4a}b-4\ge0\iff\dfrac{4a^2-4ab+b^2}{ab}\ge0$
• 4. $\dfrac ba+\dfrac{4a}b-4\ge0\iff\dfrac{4a^2-4ab+b^2}{ab}\ge0$ if and only if $ab>0$
• 5. $\dfrac{4a^2-4ab+b^2}{ab}\ge0\iff\dfrac{(2a-b)^2}{ab}\ge0$
• 6. We have $\dfrac{(2a-b)^2}{ab}\ge0$ if and only if $a>0$ and $b>0$
• 7. We have $\dfrac{(2a-b)^2}{ab}\ge0$ if $a>0$ and $b>0$
• 8. Therefore, $(a+b)\left(\dfrac1a+\dfrac4b\right)\ge9$ if $a>0$ and $b>0$
• 9. Therefore, since $a>0$ and $b>0$, $(a+b)\left(\dfrac1a+\dfrac4b\right)>9$