Inequalities | Mathematics (Maths) Class 11 - Commerce PDF Download

Inequalities

Let a and b be real numbers. If a – b is negative, we say that a is less than b (a < b) and if a – b is positive, then a is greater than b (a > b).

Important Points to be Remembered

(i) If a > b and b > c, then a > c. Generally, if a₁ > a₂, a₂ > a₃,…., a_{n – 1} > a_n, then a₁ > a_n.

(vii) If a < x < b and a, b are positive real numbers then a² < x² < b²

Important Inequality

1. Arithmetico-Geometric and Harmonic Mean Inequality

(i) If a, b > 0 and a ≠ b, then

(ii) if a_i > 0, where i = 1,2,3,…,n, then

(iii) If a₁, a₂,…, a_n are n positive real numbers and m₁, m₂,…,m_n are n positive rational numbers, then

i.e., Weighted AM > Weighted GM

(iv) If a₁, a₂,…, a_n are n positive distinct real numbers, then

(a)

(b)

(c) If a₁, a₂,…, a_n and b₁, b₂,…, b_n are rational numbers and M is a rational number, then

(d)

(v) If a₁, a₂, a₃,…, a_n are distinct positive real numbers and p, ,q, r are natural numbers, then

2. Cauchy – Schwartz’s inequality

If a₁, a₂,…, a_n and b₁, b₂,…, b_n are real numbers, such that

(a₁b₁ + a₂b₂ + …+ a_nb_n)² ≤ (a₁² + a₂² + …, a_n²) * (b₁² + b₂² + …, b_n²)

Equality holds, iff a₁ / b₁ = a₂ / b₂ = a_n / b_n

3. Tchebychef’s Inequality

Let a₁, a₂,…, a_n and b₁, b₂,…, b_n are real numbers, such that

(i) If a₁ ≤ a₂ ≤ a₃ ≤… ≤ a_n and b₁ ≤ b₂ ≤ b₃ ≤… ≤ b_n, then

n(a₁b₁ + a₂b₂ + a₃b₃ + …+ a_nb_n) ≥ (a₁ + a₂ + …+ a_n) (b₁ + b₂ + …+ b_n)

(ii) If If a₁ ≥ a₂ ≥ a₃ ≥… ≥ a_n and b₁ ≥ b₂ ≥ b₃ ≥… ≥ b_n, then

n(a₁b₁ + a₂b₂ + a₃b₃ + …+ a_nb_n) ≤ (a₁ + a₂ + …+ a_n) (b₁ + b₂ + …+ b_n)

4. Weierstrass Inequality

(i) If a₁, a₂,…, a_n are real positive numbers, then for n ≥ 2

(1 + a₁)(1 + a₂) … (1 + a_n) > 1 + a₁ + a₂ + … + a_n

(ii) If a₁, a₂,…, a_n are real positive numbers, then

(1 – a₁)(1 – a₂) … (1 – a_n) > 1 – a₁ – a₂ – … – a_n

5. Logarithm Inequality

(i) (a) When y > 1 and log_y x > z ⇒ x > y^z

(b) When y > 1 and log_y x < z ⇒ 0 < x < y^z

(ii) (a) When 0 < y < 1 and log_y x > z ⇒ 0 < x < y^z

(b) hen 0 < y < 1 and log_y x < z ⇒ x > y^z

Application of Inequalities to Find the Greatest and Least Values

(i) If x_l,x₂,…,x_n are n positive variables such that x_l + x₂ +…+ x_n = c (constant), then the product x_l * x₂ *….* x_n is greatest when x₁ = x₂ =… =x_n = c/n and the greatest value is (c/n)ⁿ.

(ii) If x_l,x₂,…,x_n are positive variables such that x_l,x₂,…,x_n = c (constant), then the sum x_l + x₂ +….+ x_n is least when x₁ = x₂ =… =x_n = c^1/n and the least value of the sum is n (c^1/n).

(iii) If x_l,x₂,…,x_n are variables and m_l,m₂,…,m_n are positive real number such that x_l + x₂ +….+ x_n = c (constant), then x_l^m_l * x₂^m₂ *… * x_n^m_n is greatest, when

x_l / m_l = x₂ / m₂ =…= x_n / m_n

= x_l + x₂ +….+ x_n / m_l + m₂ +….+ m_n