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May 19, 2024
inequalitiesalgebra

Problem Statement

(a) For non negetive a,b,c,ra,b,c, r prove that ar(a−b)(a−c)+br(b−a)(b−c)+cr(c−a)(c−b)≥0a^r(a-b)(a-c) + b^r(b-a)(b-c) + c^r (c-a)(c-b) \geq 0 (b) Find an inequality for non negative a,b,ca,b,c with a4+b4+c4+abc(a+b+c)a^4+b^4+c^4 + abc(a+b+c) on the greater side. (c) Prove that if abc=1abc = 1 for non negative a,b,ca,b,c, a4+b4+c4+a3+b3+c3+a+b+c≥a2+b2c+b2+c2a+c2+a2b+3a^4+b^4+c^4+a^3+b^3+c^3+a+b+c \geq \frac{a^2+b^2}{c}+\frac{b^2+c^2}{a}+\frac{c^2+a^2}{b}+3
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