examday

/TMUA

Let $a$ and $b$ be real numbers satisfying $a+b=2$. A student gives the following proof that $a^4+b^4\geq2$.

1. Since $(a-b)^2\geq0$, the identity $4ab=(a+b)^2-(a-b)^2$ gives $ab\leq1$.
2. Hence $a^2+b^2=(a+b)^2-2ab\geq2$.
3. Squaring the inequality in line II gives $(a^2+b^2)^2\geq4$.
4. Squaring $ab\leq1$ gives $a^2b^2\leq1$.
5. Therefore $a^4+b^4=(a^2+b^2)^2-2a^2b^2\geq4-2=2$.

Which of the following is correct?