Find the minimum of the value $\max\left(\frac{1}{ac}+b,\frac{1}{a}+bc,\frac{a}{b}+c\right)$ Let $a,b,c>0$ find the following minimum of the value
$$\max\left(\dfrac{1}{ac}+b,\dfrac{1}{a}+bc,\dfrac{a}{b}+c\right)$$
the book is hint:when $a=b=c=1$,then the minimum of the value is $2$

Hint:Use AM-GM $$\max\left(\dfrac{1}{ac}+b,\dfrac{1}{a}+bc,\dfrac{a}{b}+c\right)\ge \dfrac{\frac{1}{ac}+b+\frac{a}{b}+c}{2}\ge \dfrac{4\sqrt[4]{\frac{1}{ac}\cdot b\cdot\dfrac{a}{b}\cdot c}}{2}=2$$
