Apply inequality Cauchy, we have:

\(a+b\ge2\sqrt{ab}\)

\(\Rightarrow\left\{{}\begin{matrix}a^4+b^4\ge2\sqrt{a^4b^4}\\c^4+d^4\ge2\sqrt{c^4d^4}\end{matrix}\right.\)

\(\Leftrightarrow a^4+b^4+c^4+d^4\ge2a^2b^2+2c^2d^2=2\left[\left(ab\right)^2+\left(cd\right)^2\right]\)

But \(\left(ab\right)^2+\left(cd\right)^2\ge2\sqrt{\left(ab\right)^2\cdot\left(cd\right)^2}=2abcd\)

\(\Leftrightarrow a^4+b^4+c^4+d^4\ge2\cdot2abcd=4abcd\)

Apply inequality Cauchy, we have:

a+b≥2ab−−√

⇒{a4+b4≥2a4b4−−−−√c4+d4≥2c4d4−−−−√

⇔a4+b4+c4+d4≥2a2b2+2c2d2=2[(ab)2+(cd)2]

But (ab)2+(cd)2≥2(ab)2⋅(cd)2−−−−−−−−−√=2abcd

⇔a4+b4+c4+d4≥2⋅2abcd=4abcd

Applying Cauchy's inequality for 4 numbers , we have 

\(a^4+b^4+c^4+d^4\ge4.\sqrt[4]{a^4.b^4.c^4.d^4}=4abcd\)

Equation occur <=> a = b = c = d