We have:

\(\dfrac{ab}{c}+\dfrac{bc}{a}+\dfrac{ca}{b}=\dfrac{ab}{2c}+\dfrac{bc}{2a}+\dfrac{ca}{2b}+\dfrac{ca}{2b}+\dfrac{ab}{2c}+\dfrac{bc}{2a}\)

\(\ge2\sqrt{\dfrac{ab}{2c}\cdot\dfrac{ca}{2b}}+2\sqrt{\dfrac{bc}{2a}\cdot\dfrac{ca}{2b}}+2\sqrt{\dfrac{ab}{2c}\cdot\dfrac{bc}{2a}}=a+b+c\)

\(\Rightarrow\dfrac{ab}{c}+\dfrac{bc}{a}+\dfrac{ac}{b}\ge a+b+c\)

The "=" occurs when a = b = c

We have:

abc+bca+cab=ab2c+bc2a+ca2b+ca2b+ab2c+bc2a

≥2√ab2c⋅ca2b+2√bc2a⋅ca2b+2√ab2c⋅bc2a=a+b+c

⇒abc+bca+acb≥a+b+c

The "=" occurs when a = b = c