We have an answer to "Proving (sin2α+sin⁡αcos⁡α)sin⁡α(cos2α+sin⁡αcos⁡α)cos⁡α≤1"

vegetars8t

Answered question

2022-01-16

Proving

{(\sin^{2} α + \sin α \cos α)}^{\sin α} {(\cos^{2} α + \sin α \cos α)}^{\cos α} \leq 1

lovagwb

Beginner2022-01-17Added 50 answers

Rewrite the last line of your proof as:

a^{\frac{a}{a + b}} \cdot b^{\frac{b}{a + b}} \cdot (a + b) \leq 1

Apply the weighted AM-GM inequality you have:

L H S \leq (\frac{a}{a + b} + \frac{b}{a + b}) \cdot (a + b) = \frac{1}{a + b} \cdot (a + b) = 1 = R H S

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