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**Inequalities**

Let a and b be real numbers. If a â€“ b is negative, we say that a is less than b (a < b) and if a â€“ 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 a_{1} > a_{2}, a_{2} > a_{3},â€¦., a_{n â€“ 1} > a_{n}, then a_{1} > a_{n}.

**(vii) If a < x < b and a, b are positive real numbers then a ^{2} < x^{2} < b^{2}**

**Important Inequality**

1. **Arithmetico-Geometric and Harmonic Mean Inequality**

(i) If a, b > 0 and a â‰ b, then

(ii) if a_{i} > 0, where i = 1,2,3,â€¦,n, then

(iii) If a_{1}, a_{2},â€¦, a_{n} are n positive real numbers and m_{1}, m_{2},â€¦,m_{n} are n positive rational numbers, then

i.e., Weighted AM > Weighted GM

(iv) If a_{1}, a_{2},â€¦, a_{n} are n positive distinct real numbers, then

(a)

(b)

(c) If a_{1}, a_{2},â€¦, a_{n} and b_{1}, b_{2},â€¦, b_{n} are rational numbers and M is a rational number, then

(d)

(v) If a_{1}, a_{2}, a_{3},â€¦, a_{n} are distinct positive real numbers and p, ,q, r are natural numbers, then

2. **Cauchy â€“ Schwartzâ€™s inequality**

If a_{1}, a_{2},â€¦, a_{n} and b_{1}, b_{2},â€¦, b_{n} are real numbers, such that

(a_{1}b_{1} + a_{2}b_{2} + â€¦+ a_{n}b_{n})^{2} â‰¤ (a_{1}^{2} + a_{2}^{2} + â€¦, a_{n}^{2}) * (b_{1}^{2} + b_{2}^{2} + â€¦, b_{n}^{2})

Equality holds, iff a_{1} / b_{1} = a_{2} / b_{2} = a_{n} / b_{n}

3. **Tchebychefâ€™s Inequality**

Let a_{1}, a_{2},â€¦, a_{n} and b_{1}, b_{2},â€¦, b_{n} are real numbers, such that

(i) If a_{1} â‰¤ a_{2} â‰¤ a_{3} â‰¤â€¦ â‰¤ a_{n} and b_{1} â‰¤ b_{2} â‰¤ b_{3} â‰¤â€¦ â‰¤ b_{n}, then

n(a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3} + â€¦+ a_{n}b_{n}) â‰¥ (a_{1} + a_{2} + â€¦+ a_{n}) (b_{1} + b_{2} + â€¦+ b_{n})

(ii) If If a_{1} â‰¥ a_{2} â‰¥ a_{3} â‰¥â€¦ â‰¥ a_{n} and b_{1} â‰¥ b_{2} â‰¥ b_{3} â‰¥â€¦ â‰¥ b_{n}, then

n(a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3} + â€¦+ a_{n}b_{n}) â‰¤ (a_{1} + a_{2} + â€¦+ a_{n}) (b_{1} + b_{2} + â€¦+ b_{n})

4. **Weierstrass Inequality**

(i) If a_{1}, a_{2},â€¦, a_{n} are real positive numbers, then for n â‰¥ 2

(1 + a_{1})(1 + a_{2}) â€¦ (1 + a_{n}) > 1 + a_{1} + a_{2} + â€¦ + a_{n}

(ii) If a_{1}, a_{2},â€¦, a_{n} are real positive numbers, then

(1 â€“ a_{1})(1 â€“ a_{2}) â€¦ (1 â€“ a_{n}) > 1 â€“ a_{1} â€“ a_{2} â€“ â€¦ â€“ a_{n}

5. **Logarithm Inequality**

(i) (a) When y > 1 and log_{y} x > z â‡’ x > y^{z}

(b) When y > 1 and log_{y} x < z â‡’ 0 < x < y^{z}

(ii) (a) When 0 < y < 1 and log_{y} x > z â‡’ 0 < x < y^{z}

(b) hen 0 < y < 1 and log_{y} x < z â‡’ x > y^{z}

**Application of Inequalities to Find the Greatest and Least Values**

(i) If x_{l},x_{2},â€¦,x_{n} are n positive variables such that x_{l} + x_{2} +â€¦+ x_{n} = c (constant), then the product x_{l} * x_{2} *â€¦.* x_{n} is greatest when x_{1} = x_{2} =â€¦ =x_{n} = c/n and the greatest value is (c/n)^{n}.

(ii) If x_{l},x_{2},â€¦,x_{n} are positive variables such that x_{l},x_{2},â€¦,x_{n} = c (constant), then the sum x_{l} + x_{2} +â€¦.+ x_{n} is least when x_{1} = x_{2} =â€¦ =x_{n} = c^{1/n} and the least value of the sum is n (c^{1/n}).

(iii) If x_{l},x_{2},â€¦,x_{n} are variables and m_{l},m_{2},â€¦,m_{n} are positive real number such that x_{l} + x_{2} +â€¦.+ x_{n} = c (constant), then x_{l}^{m}_{l} * x_{2}^{m}_{2} *â€¦ * x_{n}^{m}_{n} is greatest, when

x_{l} / m_{l} = x_{2} / m_{2} =â€¦= x_{n} / m_{n}

= x_{l} + x_{2} +â€¦.+ x_{n} / m_{l} + m_{2} +â€¦.+ m_{n}

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