[Math] Show: $(x+y)^4 \leq 8(x^4 + y^4)$

Question

I wish to show the following statement:

$
\forall x,y \in \mathbb{R}
$

$$
(x+y)^4 \leq 8(x^4 + y^4)
$$

What is the scope for generalisaion?

Edit:

Apparently the above inequality can be shown using the Cauchy-Schwarz inequality. Could someone please elaborate, stating the vectors you are using in the Cauchy-Schwarz inequality:

$\ \ \forall \ \ v,w \in V, $ an inner product space,

$$|\langle v,w\rangle|^2 \leq \langle v,v \rangle \cdot \langle w,w \rangle$$

where $\langle v,w\rangle$ is an inner product.

Answer 1

Apply Cauchy-Schwarz inequality twice: $x^4 + y^4 \geq \dfrac{1}{2}\left(x^2+y^2\right)^2 \geq \dfrac{1}{2}\left(\dfrac{1}{2}\left(x+y\right)^2\right)^2 = \dfrac{1}{8}\left(x+y\right)^4$.