<s>[INST] $(a^4+b^2) \leq (a^2+b^2)^2$ for all $a, b \in \mathbb{R}_+$? Let $a, b \in \mathbb{R}_+$.
Question. Is valid that
$$(a^4+b^2) \leq (a^2+b^2)^2?\tag{1}$$
I tried what follows:
$$(a^2+b^2)^2=a^4+2a^2b^2+b^4 \geq a^4+2a^2b^2.$$
For this, can I to conclude that $(1)$ holds?
 [/INST] HINT
You can simply expand the RHS and subtract the LHS:
\begin{align*}
(a^{2} + b^{2})^{2} = a^{4} + 2a^{2}b^{2} + b^{4} \Rightarrow (a^{2} + b^{2})^{2} - a^{4} - b^{2} = 2a^{2}b^{2} + b^{4} - b^{2} = b^{2}(2a^{2} + b^{2} - 1)
\end{align*}
When the last expression is nonnegative?

 Answer: outside the ellipse $2a^{2} + b^{2} = 1$

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