gaokao 2015 Q14

gaokao · China · tianjin-science 5 marks Vectors Introduction & 2D Optimization of a Vector Expression
In isosceles trapezoid ABCD, $\mathrm{AB} \parallel \mathrm{DC}$, $\mathrm{AB} = 2$, $\mathrm{BC} = 1$, $\angle \mathrm{ABC} = 60°$. Moving points E and F are on segments BC and DC respectively, with $\overrightarrow{\mathrm{BE}} = \lambda\overrightarrow{\mathrm{BC}}$, $\overrightarrow{\mathrm{DF}} = \frac{1}{9\lambda}\overrightarrow{\mathrm{DC}}$. Then the minimum value of $\overrightarrow{\mathrm{AE}} \cdot \overrightarrow{\mathrm{AF}}$ is . 
In isosceles trapezoid ABCD, $\mathrm{AB} \parallel \mathrm{DC}$, $\mathrm{AB} = 2$, $\mathrm{BC} = 1$, $\angle \mathrm{ABC} = 60°$. Moving points E and F are on segments BC and DC respectively, with $\overrightarrow{\mathrm{BE}} = \lambda\overrightarrow{\mathrm{BC}}$, $\overrightarrow{\mathrm{DF}} = \frac{1}{9\lambda}\overrightarrow{\mathrm{DC}}$. Then the minimum value of $\overrightarrow{\mathrm{AE}} \cdot \overrightarrow{\mathrm{AF}}$ is \underline{\hspace{2cm}}.
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