<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] $$\bigg( \left(\frac{1}{2} \right)^2+ \left(\frac{1}{2} \right)^2 \bigg)^2=\left(\frac{1}{2}\right)^2=\frac{1}{4}$$
but
$$\bigg( \left(\frac{1}{2}\right)^4+ \left(\frac{1}{2}\right)\bigg)^2= \left(\frac{1}{2}\right)^4+\frac{1}{4} > \frac{1}{4}= \bigg( \left(\frac{1}{2} \right)^2+ \left(\frac{1}{2}\right)^2 \bigg)^2$$
</s>