Question-148364

Question Number 148364 by mathdanisur last updated on 27/Jul/21

Commented by dumitrel last updated on 27/Jul/21

\Leftrightarrow {({a e}^{\frac{x}{2}} + {b e}^{\frac{y}{2}})}^{2} ⩾ e^{\frac{a x + b y}{a + b}} \Leftrightarrow

{a e}^{\frac{x}{2}} + {b e}^{\frac{y}{2}} ⩾ e^{\frac{a x + b y}{2 (a + b)}}

f (x) = e^{\frac{x}{2}} c o n v e x a; \frac{a}{a + b} + \frac{b}{a + b} = 1 \Rightarrow

\frac{a}{a + b} e^{\frac{x}{2}} + \frac{b}{a + b} e^{\frac{y}{2}} ⩾ c^{\frac{a}{a + b} \cdot \frac{x}{2} + \frac{b}{a + b} \cdot \frac{y}{2}} = e^{\frac{a x + b y}{2 (a + b)}}

Commented by mathdanisur last updated on 27/Jul/21

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