<s>[INST] If $a, b, c >0$ prove that $ [(1+a)(1+b)(1+c)]^7 > 7^7a^4b^4c^4 $. I solved it using AM, GM inequalities and reached to $[(1+a)(1+b)(1+c)]^7 > 2^{21}(abc)^\frac72 $ please help how to get $7^7(abc)^4$ in the inequality.
 [/INST] We only need to prove that for positive $x$ we have $(1+x)^7\gt 7^{7/3}x^4$. 
For positive $x$, let 
$$f(x)=\frac{(1+x)^7}{x^4}.$$
By using $f'(x)$ we find that $f(x)$ reaches a minimum at $x=4/3$. The minimum value of $f(x)$ is 
$$\frac{7^7}{3^34^4,}$$
which is greater than $7^{7/3}$. 
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