How to show that $a^3+b^3+c^3+d^3\geq abc+abd+acd+bcd$ if $a,b,c,d>0$

Question

How can I prove that if $a,b,c,d>0$ then

$$a^3+b^3+c^3+d^3\geq abc+abd+acd+bcd?$$

I think there is some simple proof but I can't remember... is this a special case of some general inequality?

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I need it to prove that if $a^3+2c=3ab$ then all of 4 positive roots of following polynomial is the same.

$x^4+ax^3+bx^2+cx+d$

Answer 1

hint: Use AM-GM inequality $4$ times:

$$x^3+y^3+z^3 \geq 3xyz$$

with $(x,y,z) = (a,b,c), (a,c,d), (a,b,d), (b,c,d)$, and add up.