Summary of Inequalities 1 | CAT Preparation 2024 | Algebra | Quantitative Aptitude

Summary of "Inequalities 1 | CAT Preparation 2024 | Algebra | Quantitative Aptitude"

This video, presented by Ravi Prakash, covers fundamental concepts and rules related to inequalities, especially focusing on maximizing or minimizing expressions involving sums and products of variables under certain constraints. The content is tailored for CAT exam preparation and introduces several important inequality principles, often derived from the Arithmetic Mean-Geometric Mean (AM-GM) inequality, with practical examples and applications.

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Main Ideas and Concepts

1. Basic Principle: Product Maximization When Sum is Constant
   • If the sum of variables is constant, their product is maximized when the variables are equal or as close as possible.
   • Example: For \( a + b = 36 \), the product \( a \times b \) is maximum when \( a = b = 18 \).
   • This applies to natural numbers and can be extended to more variables.
2. Sum Minimization When Product is Constant
   • If the product of variables is constant, their sum is minimized when the variables are equal or as close as possible.
   • Example: For \( a \times b = 324 \), the minimum sum \( a + b \) occurs when \( a = b = 18 \).
   • If the product is not a perfect square (e.g., 132), choose natural numbers closest to the square root (e.g., 11 and 12 for 132), and sum is minimized accordingly.
3. Maximizing Expressions of the Form \( a^m \times b^n \) with Constant Sum
   • If \( a + b \) is constant, then \( a^m \times b^n \) is maximized when \( \frac{a}{m} = \frac{b}{n} \).
   • Application: Maximizing the volume of a cylinder with fixed sum of radius and height \( (R + H = \text{constant}) \).
   • Volume \( V = \pi R^2 H \) is maximized when \( \frac{R}{2} = \frac{H}{1} \), i.e., \( R:H = 2:1 \).
4. Minimizing Sum When \( a^m \times b^n \) is Constant
   • The sum \( a + b \) is minimized when \( \frac{a}{m} = \frac{b}{n} \), given that \( a^m \times b^n \) is constant.
5. Generalization with Coefficients: \( k_1 a + k_2 b = \text{constant} \)
   • When the sum involves coefficients, the maximum of \( a^m \times b^n \) occurs when \( \frac{k_1 a}{m} = \frac{k_2 b}{n} \).
   • This extends the third point to weighted sums.
6. Minimizing Weighted Sums When \( a^m \times b^n \) is Constant
   • Similarly, when \( a^m \times b^n \) is constant, the weighted sum \( k_1 a + k_2 b \) is minimized when \( \frac{k_1 a}{m} = \frac{k_2 b}{n} \).

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Methodology / Key Rules (Bullet Points)

• Rule 1:
  If \( a + b = \text{constant} \), then \( a \times b \) is maximum when \( a = b \) or as close as possible.
• Rule 2:
  If \( a \times b = \text{constant} \), then \( a + b \) is minimum when \( a = b \) or as close as possible.
• Rule 3:
  For constants \( m, n \), if \( a + b = \text{constant} \), then \( a^m \times b^n \) is maximum when:
  \(\frac{a}{m} = \frac{b}{n}\)
• Rule 4:
  If \( a^m \times b^n = \text{constant} \), then \( a + b \) is minimum when:
  \(\frac{a}{m} = \frac{b}{n}\)
• Rule 5:
  If \( k_1 a + k_2 b = \text{constant} \), then \( a^m \times b^n \) is maximum when:
  \(\frac{k_1 a}{m} = \frac{k_2 b}{n}\)
• Rule 6:
  If \( a^m \times b^n = \text{constant} \), then the weighted sum \( k_1 a + k_2 b \) is minimized when:
  \(\frac{k_1 a}{m} = \frac{k_2 b}{n}\)

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Educational

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