If a, b, c are real and (bc+ca+ab)^2>k(a+b+c), then k equals (a) abc (b) 3 (c) 1 (d) 3 abc

Name: Problem 110, Chapter 5: IIT-JEE Super Course in Mathematics
Uploaded: 2021-09-09T07:08:03-08:00
Duration: 4 min 17 s
Channel: Gaurav Kalra
Description: If a, b, c are real and (bc+ca+ab)^2>k(a+b+c), then k equals (a) abc (b) 3 (c) 1 (d) 3 abc

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If a, b, c are real and (bc+ca+ab)^2>k(a+b+c), then k equals...

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are real and $(\mathrm{bc}+\mathrm{ca}+\mathrm{ab})^{2}>\mathrm{k}(\mathrm{a}+\mathrm{b}+\mathrm{c})$, then $\mathrm{k}$ equals
(a) $\mathrm{abc}$
(b) 3
(c) 1
(d) $3 \mathrm{abc}$

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Key Concepts

Equality Conditions and Test Cases

Determining when an inequality is tight—that is, when equality is achieved—is a powerful strategy in inequality analysis. Testing particular cases, such as when all variables are equal or when some take on extreme values, can help isolate critical values of parameters or constants in the inequality and verify the optimal conditions under which the inequality holds.

Algebraic Inequalities

Algebraic inequalities involve finding relationships of order or bounds between different algebraic expressions. Analyzing such inequalities often requires creative manipulations, including the use of symmetric functions, testing special cases, and applying standard techniques like homogenization or substitution to compare the degrees of expressions or deduce optimal constants.

Symmetric Functions

Symmetric functions are expressions in multiple variables that remain invariant under any permutation of the variables. In algebra, elementary symmetric sums—such as the sum of the variables, the sum of their pairwise products, and the product of the variables—play a central role in understanding the relationships between the roots of polynomials and in reformulating equations and inequalities in a more tractable, invariant form.

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