Find the minimum value of $P=ab+bc+3ca+\dfrac{3}{a+b+c}$ Let $a,b,c$ be non-negative real numbers such that $a^2+b^2+c^2=3$. Find the minimum value of $$P=ab+bc+3ca+\dfrac{3}{a+b+c}.$$
This is an asymmetric inequality. It is hard for me to find when the equation holds. I guess when it occurs if $a=c=0$ and $b=\sqrt{3}$. Then $\min P=\sqrt{3}$. But I have no idea to solve it. Please help me a hint. Thank you.

No matter what, by simply calculating one has $P(\sqrt 3, 0, 0) = \sqrt 3$. On the other hand,

We know a minimum must exist by Weierstrass's theorem. This is because $Z := \{(a, b, c) \in \mathbb R ^3\mid a, b, c \geq 0, a^2 + b^2 + c^2 = 3\}$ is a closed bounded set and $P$ is continuous on $Z$.

Write $$P(a,b,c) = S_1 + S_2 \tag{1}$$
where $S_1 :=ab + bc + \dfrac{3}{2} + ac + \dfrac{3}{a + b + c}$ and $S_2 :=2ac - \dfrac{3}{2}$. Note that $S_1$ is symmetric. The choice of $\dfrac{3}{2}$ will be clear should you read on.

If it is possible to simultaneously minimise $S_1$ and $S_2$ (with given constraints), then their sum will be the minimum value of $P$. Let us try to minimise $S_1$.

Plugging $bc = \dfrac{1}{2}(b+c)^2 - \dfrac{3}{2} + \dfrac{a^2}{2}$ into $S_1$ yields
$$ S_1 = \dfrac{1}{2}(b+c)^2 + (b + c) a + \frac{3}{(b + c) + a} + \dfrac{a^2}{2},$$
or with $x := b + c$,
\begin{align}
\begin{split}
S_1(x, a) &= \dfrac{1}{2} x^2 + a\, x + \frac{3}{x + a} + \dfrac{a^2}{2} = \\
 &= \dfrac{1}{2} (x + a)^2 + \dfrac{3}{x + a} = \\
 &= \dfrac{1}{2} z^2 + \dfrac{3}{z}.
\end{split}
\tag{2}
\end{align}

*

*We are left with the problem of minimising $S_1(z)$ wrt $z$ when $z \geq \sqrt 3$ (because $z^2 \geq a^2 + b^2 + c^2$).


*The derivative with respect to $z$ is $z - \dfrac{3}{z^2}$. This is positive if $z > \sqrt[3]{3}$. So $S_1$ is increasing for $z \geq \sqrt[3]{3}$ and we should pick $z_0 := \sqrt 3$ if we can (choosing $z_0 := \sqrt[3]{3}$ is not possible because $\sqrt[3]{3} < \sqrt 3$).
If one moves up the chain, we have $z_0 = x_0 + a_0 = a_0 + b_0 + c_0 = \sqrt 3$. And we see it is possible to choose $a_0 := \sqrt 3$ and $b_0 := c_0 := 0$ to obtain $z_0 = \sqrt 3$, thus minimising $S_1$. By picking $c_0 = 0$, $S_2$ is minimised as well, so we are done. In conclusion, the minimum value of $P$, subject to given constraints, is $\sqrt 3$. It is obtained at, for example, $(a_0, b_0, c_0) = (\sqrt 3, 0, 0)$.
