If a, b, c are distinct and positive, and (2 a+3 b+5 c)^10>λa^2 b^3 c^5, then λ= (a) 10 (b) 10

step 1 In this problem I will apply AM is greater than GM inequality. So I just applying AM is greater

If a, b, c are distinct and positive, and (2 a+3 b+5...

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are distinct and positive, and $(2 \mathrm{a}+3 \mathrm{~b}+5 \mathrm{c})^{10}>\lambda \mathrm{a}^{2} \mathrm{~b}^{3} c^{5}$, then $\lambda=$
(a) 10
(b) $10^{10}$
(c) $10^{\prime / 2}$
(d) $\frac{1}{10}$

Key Concepts

Equality Conditions in Inequalities

Understanding the conditions under which an inequality becomes an equality is key to solving optimization problems. For the AM-GM inequality, this occurs when the weighted terms are proportional to one another, which helps in determining the optimal relationship among variables in an expression. Recognizing these conditions allows for precise determination of parameters or constants that maximize or minimize the expression.

Homogeneity of Expressions

Homogeneous expressions, where every term is of the same total degree, allow for variable scaling and normalization. This property is particularly useful when applying inequalities like AM-GM since it permits a reduction in the number of independent variables, making it easier to identify optimal configurations and determine sharp bounds or constants within the problem context.

Weighted Arithmetic Mean-Geometric Mean (AM-GM) Inequality

The weighted AM-GM inequality generalizes the standard AM-GM inequality by incorporating weights for each term, stating that a weighted arithmetic mean is always at least as large as the corresponding weighted geometric mean for non-negative numbers. This principle is crucial in problems where different terms contribute unequally, allowing one to establish bounds or optimize expressions by equating the terms to achieve equality in the inequality.

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