Let a- b- c - 0- a-b-c-15-The least value of E-a3-a2-ab-b2-b3-b2-bc-c2-c3-c2-ca-a2 is

Observe that a^3 b^3/a^2+ab+b^2+b^3 c^3/b^2+bc+c^2+c^3 a^3/c^2+ca+a^2 =(a b)+(b c)+(c a)=0 Thus, we can write E=1/2 ( a^3+b^3/a^2+ab+b^2+b^3+c^3/b^2+bc+c ...

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Let a, b, c >...

Question

Let $a, b, c > 0, a + b + c = 15$ .
The least value of $E = \frac{a^{3}}{a^{2} + a b + b^{2}} + \frac{b^{3}}{b^{2} + b c + c^{2}} + \frac{c^{3}}{c^{2} + c a + a^{2}}$ is___

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Solution

Observe that
$\frac{a^{3} - b^{3}}{a^{2} + a b + b^{2}} + \frac{b^{3} - c^{3}}{b^{2} + b c + c^{2}} + \frac{c^{3} - a^{3}}{c^{2} + c a + a^{2}} = (a - b) + (b - c) + (c - a) = 0 T h u s, w e c a n w r i t e E = \frac{1}{2} (\frac{a^{3} + b^{3}}{a^{2} + a b + b^{2}} + \frac{b^{3} + c^{3}}{b^{2} + b c + c^{2}} + \frac{c^{3} + a^{3}}{c^{2} + c a + a^{2}}) B u t \frac{a^{3} + b^{3}}{a^{2} + a b + b^{2}} \geq \frac{a + b}{3} S i n c e a^{2} - 2 a b + b^{2} = (a - b)^{2} \geq 0 ∴ E \geq \frac{1}{3} (a + b + c) = 5$

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Let a, b, c>0, a+b+c=15. The least value of E=a3a2+ab+b2+b3b2+bc+c2+c3c2+ca+a2 is___

Observe that a3−b3a2+ab+b2+b3−c3b2+bc+c2+c3−a3c2+ca+a2=(a−b)+(b−c)+(c−a)=0Thus, we can writeE=12(a3+b3a2+ab+b2+b3+c3b2+bc+c2+c3+a3c2+ca+a2)But a3+b3a2+ab+b2≥a+b3Since a2−2ab+b2=(a−b)2≥0∴E≥13(a+b+c)=5

Let $a, b, c > 0, a + b + c = 15$ .
The least value of $E = \frac{a^{3}}{a^{2} + a b + b^{2}} + \frac{b^{3}}{b^{2} + b c + c^{2}} + \frac{c^{3}}{c^{2} + c a + a^{2}}$ is___