Problem 8 - Algebra for JEE ADVANCED

Question

The minimum value of \(P=b c x+c a y+a b z\), when \(x y z=a b c\), is a. \(3 a b c\) b. \(6 a b c\) c. \(a b c\) d. \(4 a b c\)

Step-by-Step Solution

Answer

The minimum value of \(P\) is \(3abc\), option (a).

Step 1: Understanding the Problem

We are given the expression \(P=b c x+c a y+a b z\) with the constraint \(xyz=abc\). The task is to find the minimum value of \(P\) given this constraint.

Step 2: Approach Using AM-GM Inequality

The Arithmetic Mean-Geometric Mean (AM-GM) inequality states that for non-negative numbers, the arithmetic mean is greater than or equal to the geometric mean. We'll apply it to the terms \(bx\), \(cy\), and \(az\) from the expression \(P\).

Step 3: Apply AM-GM to the Terms

By applying AM-GM to \(bx\), \(cy\), \(az\), we have:\[\frac{bcx + cay + abz}{3} \geq \sqrt[3]{(bcx)(cay)(abz)} = \sqrt[3]{abc \, \cdot \, bxa \, \cdot \, cyz \, \cdot \, abz} = \sqrt[3]{(abc)^3} = abc\]

Step 4: Solve the Inequality for P

From the AM-GM inequality, we have:\[bcx + cay + abz \geq 3abc\]Thus, the minimum value for \(P\) using AM-GM inequality is \(3abc\).

Step 5: Verify the Minimum Condition

The equality holds in the AM-GM inequality when \(b x = c y = a z\). This implies \(x = \frac{abc}{a^2} = b, y = \frac{abc}{b^2} = c, z = \frac{abc}{c^2} = a\). By substitution, the constraint \(xyz = abc\) holds, confirming that \(3abc\) is attainable.

Step 6: Conclusion

Therefore, the minimum value of the given expression \(P\) is \(3abc\), which corresponds to option (a).

Key Concepts

Minimum Value Problems

Minimum value problems are a common topic in mathematics, especially in optimization and calculus. They involve finding the lowest point or value of a function given certain constraints. In our exercise, we are tasked with determining the minimum value of \( P = bc x + ca y + ab z \). Here, the constraint is that the product \( xyz = abc \) must hold.

• Understanding constraints is crucial. It restricts possible solutions and guides us towards the minimum value.
• Using inequalities like the AM-GM inequality helps in establishing bounds or minimums of expressions.
• Verification is key; once a potential minimum is found, check against the constraints to ensure it is achievable.

These steps ensure we find the correct solution while adhering to given conditions.

Algebra

Algebra forms the backbone of solving expressions and equations, such as in this problem. By manipulating expressions, we can simplify and solve complex minimum value problems.
In this case, algebraic manipulation is required to apply the AM-GM inequality appropriately and successfully. Treat the terms \( bcx, cay, \) and \( abz \) as linked products, each contributing to the overall minimum value of the expression.

• Symmetry in algebra helps us find equilibrium. Upon equal distribution, influencing terms like in the AM-GM are minimized effectively.
• Breaking down variables to their simplest forms can simplify initial constraints and help manage the problem better.
• Ensure thorough understanding of terms - here \( bx, cy, \text{and} az \) must be handled with care to harness the correct application of algebraic theorems.

This meticulous approach aids in uncovering the minimum value of algebraic expressions efficiently.

JEE Advanced Mathematics

JEE Advanced Mathematics is known for its challenging problems, which often demand a deep understanding of mathematical concepts. Minimum value problems like the one in this exercise are common and test a student's ability to apply inequalities, algebraic manipulation, and logical reasoning.
What makes these problems particularly challenging is the layered simplicity. The base structures are often simple inequalities, yet they hold complex relationships demanding rigorous analysis.

• Leveraging known inequalities (such as AM-GM) reflects both theoretical understanding and practical application skills.
• Devising a path from given equations to claimed results involves strategic planning and stepwise progression.
• The success in such exams hinges on the ability to understand conditions and constraints deeply, beyond just formulaic knowledge.

With these skills, students can tackle even the toughest problems in JEE Advanced Maths, gaining insights into more profound mathematical realms.