Solution:
Note that
$$\left(x^{2}+y^{2}+z^{2}\right)^{2}=x^{4}+y^{4}+z^{4}+2 x^{2} y^{2}+2 x^{2} z^{2}+2 y^{2} z^{2}$$
Using the AM-GM Inequality, we find that
$$2 x^{2} y^{2}+2 x^{2} z^{2}+2 y^{2} z^{2} \leq 2\left[\left(x^{4}+y^{4}\right) / 2\right]+2\left[\left(x^{4}+z^{4}\right) / 2\right]+2\left[\left(y^{4}+z^{4}\right) / 2\right]$$
which when combined with the previous equation results to
$$\left(x^{2}+y^{2}+z^{2}\right)^{2} \leq 3\left(x^{4}+y^{4}+z^{4}\right)$$
Therefore $k=3$.