Code:
$ a^{x_{1}}b^{x_{2}}+a^{x_{2}}b^{x_{1}}=(w_{1}a^{x_{1}}b^{x_{2}}+w_{2}a^{x_{2}}b^{x_{1}})+(w_{2}a^{x_{1}}b^{x_{2}}+w_{1}a^{x_{2}}b^{x_{1}})\geq a^{w_{1}x_{1}+w_{2}x_{2}}b^{w_{1}x_{2}+w_{2}x_{1}}+a^{w_{2}x_{1}+w_{1}x_{2}}b^{w_{1}x_{1}+w_{2}x_{2}}=a^{y_{1}}b^{y_{2}}+a^{y_{2}}b^{y_{1}} $