<s>[INST] Show that $(a^3+a+1)(b^3+b+1)(c^3+c+1)\le 27$ Let $a,b,c\ge 0$ be such that $a^2+b^2+c^2=3$. Show that
$$(a^3+a+1)(b^3+b+1)(c^3+c+1)\le 27$$
I want to consider the function
$$f(x)=\ln{(x^{3/2}+x^{1/2}+1)}$$ Maybe it isn't the case $f''(x)\le 0$, so I can't use Jensen's inequality.
 [/INST] Let $f,g$ be given by
$$
\left\lbrace
\begin{align*}
f(a,b,c)&=(a^3+a+1)(b^3+b+1)(c^3+c+1)\\[4pt]
g(a,b,c)&=a^2+b^2+c^2\\[4pt]
\end{align*}
\right.
$$
Using the method of Lagrange Multipliers, if a point $(a,b,c)$ satisfying the constraint $g(a,b,c)=3$ is such that $f$ has a local extremum at $(a,b,c)$, then for some $\lambda\in\mathbb{R}$ we must have
$$
\left\lbrace
\begin{align*}
(3a^2+1)(b^3+b+1)(c^3+c+1)&=2\lambda a&&(\text{eq}1)\\[4pt]
(a^3+a+1)(3b^2+1)(c^3+c+1)&=2\lambda b&&(\text{eq}2)\\[4pt]
(a^3+a+1)(b^3+b+1)(3c^2+1)&=2\lambda c&&(\text{eq}3)\\[4pt]
a^2+b^2+c^2&=3&&(\text{eq}4)\\[4pt]
\end{align*}
\right.
$$
Given $a,b,c\ge 0$, each of equations $1,2,3$ has positive $\text{LHS}$, so must also have positive $\text{RHS}$, hence $a,b,c,\lambda > 0$.

Without loss of generality, assume $a=\max(a,b,c)$.

Then from $(\text{eq}4)$, it follows that $1\le a < \sqrt{3}$.

Suppose $a\ne b$.

Dividing $(\text{eq}1)$ by $(\text{eq}2)$, we get
\begin{align*}
&
\frac
{(3a^2+1)(b^3+b+1)}
{(3b^2+1)(a^3+a+1)}
=
\frac{a}{b}
\\[4pt]
\implies\; &
a(a^3+a+1)(3b^2+1)=b(b^3+b+1)(3a^2+1)
\\[4pt]
\implies\; &
a(a^3+a+1)(3b^2+1)-b(b^3+b+1)(3a^2+1)=0
\\[4pt]
\implies\; &
(a-b)(3a^3b^2+3a^2b^3+a^3+b^3+a^2b+ab^2-3ab+a+b+1)=0
\\[4pt]
\implies\; &
3a^3b^2+3a^2b^3+a^3+b^3+a^2b+ab^2-3ab+a+b+1=0
\\[4pt]
\implies\; &
3a^3b^2+3a^2b^3+a^3+b^3+a^2b+ab^2+a+b=3ab-1
\\[4pt]
\implies\; &
(a+b)(3a^2b^2+a^2+b^2+1)=3ab-1
\\[4pt]
\implies\; &
(b+1)(4b^2+2)< 6b-1
\quad\text{[replaced $a$ by $1$ on the left and by $2$ on the right]}
\\[4pt]
\implies\; &
4b^3+4b^2-4b+3 < 0
\\[4pt]
\implies\; &
(4b^2-4b+1)+(4b^3+2) < 0
\\[4pt]
\implies\; &
(2b-1)^2+(4b^3+2) < 0
\\[4pt]
\end{align*}
contradiction.$\;$Hence $a=b$.

Analogously we get $a=c$.

Then $a=b=c$, hence from $(\text{eq}4)$, we get $a=b=c=1$.

It follows that for $a,b,c\ge 0$, the maximum value of $f(a,b,c)$ subject to the constraint $g(a,b,c)=3$ is $f(1,1,1)=27$.
</s>