Problem

請寫出計算及證明過程，勿只寫答案。

4. Fix positive numbers $a,b,c$. For any $(x,y,z)$ on the ellipsoid $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1$ with $x>0$, $y>0$, $z>0$, let $A(x,y,z)$, $B(x,y,z)$ and $C(x,y,z)$ be the intercepts that the tangent plane of the ellipsoid at $(x,y,z)$ makes on the $x$-axis, the $y$-axis and the $z$-axis, respectively. Find the minimum of $A(x,y,z)+B(x,y,z)+C(x,y,z)$.