<s>[INST] Prove this $4(a+b+c+d)+(a^3+b^3+c^3+d^3)\le 20$ let $a,b,c,d\in R$,and such $a^2+b^2+c^2+d^2=4$, prove or disprove
$$4(a+b+c+d)+(a^3+b^3+c^3+d^3)\le 20$$
I try use Cauchy-Schwarz inequality have
$$4(a+b+c+d)\le 4\sqrt{4(a^2+b^2+c^2+d^2)}=16$$
but
$$(a^3+b^3+c^3+d^3)^2(1+1+1+1)\ge (a^2+b^2+c^2+d^2)^3$$
so we have
$$a^3+b^3+c^3+d^3\ge 4$$
 [/INST] You found $a^3+b^3+c^3+d^3\geq 4$
Now using Tchebychev inequality we have:
$$(a+b+c+d)(a^2+b^2+c^2+d^2)\leq 4(a^3+b^3+c^3+d^3)$$
Or:
$$4(a+b+c+d)\leq 4(a^3+b^3+c^3+d^3)$$
$$4(a+b+c+d)+(a^3+b^3+c^3+d^3)\leq 5(a^3+b^3+c^3+d^3)$$
But:
$$5(a^3+b^3+c^3+d^3)\geq 20$$
therefore:
$$4(a+b+c+d)+(a^3+b^3+c^3+d^3)\leq 20$$
</s>