prove that : $\dfrac{a^2}{2}+\dfrac{b^3}{3}+\dfrac{c^6}{6} \geq abc$ for $a ,b ,c \in \mathbb{R}^{>0}$ prove that : $\dfrac{a^2}{2}+\dfrac{b^3}{3}+\dfrac{c^6}{6} \geq abc$ for $a ,b ,c \in \mathbb{R}^{>0}$ .
I think that must I use from $\dfrac{a^2}{2}+\dfrac{b^2}{2} \geq ab$ but no result please help me .!

In general, if
$$\sum_{i=1}^n \frac{1}{p_i} = 1$$
for positive $p_i$, then Jensen's inequality implies that for any concave function $\varphi$ defined on $\mathbb{R}^+$, we have
$$\varphi\left(\sum_{i=1}^n \frac{x_i^{p_i}}{p_i}\right) \ge \sum_{i=1}^n \frac{1}{p_i} \varphi(x_i^{p_i}),$$
for positive $x_i$.  In particular, if $\varphi(x) = \log x$, then we obtain Young's inequality:
$$\sum_{i=1}^n \frac{x_i^{p_i}}{p_i} \ge \prod_{i=1}^n x_i.$$
