If a > 0, b > 0 and c > 0, prove that...

MathematicsRelation Between A.M., G.M., and H.M.JEE Advanced 1984Easy

If $a > 0, b > 0$ and $c > 0$ , prove that $(a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \geq 9$ .

Visualized Solution (Hindi)

Introduction to the Problem

Given: $a, b, c > 0$
To Prove: $(a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \geq 9$

The .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A M \geq H M Tool

Concept: For positive real numbers, Arithmetic Mean (AM) $\geq$ Harmonic Mean (HM).
This inequality holds for any $n$ positive numbers.

Defining Arithmetic Mean (.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A M)

For three numbers $a, b, c$ :
$A M = \frac{a + b + c}{3}$

Defining Harmonic Mean (.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } H M)

For three numbers $a, b, c$ :
$H M = \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}$

Applying .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A M \geq H M Inequality

Substitute $A M$ and $H M$ into the inequality:
$\frac{a + b + c}{3} \geq \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}$

Final Rearrangement

Multiply both sides by $3$ and by $(\frac{1}{a} + \frac{1}{b} + \frac{1}{c})$ :
$(a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \geq 3 \times 3$
$(a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \geq 9$

Conclusion and Key Takeaway

Final Result: $(a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \geq 9$
Equality Condition: $a = b = c$
Generalization: For $n$ positive numbers, $(\sum a_{i}) (\sum \frac{1}{a _{i}}) \geq n^{2}$

00:00 / 00:00

Conceptually Similar Problems

MathematicsRelation Between A.M., G.M., and H.M.JEE Advanced 2004Moderate

View in:English Hindi

If $a, b, c$ are positive real numbers. Then prove that $(a + 1)^{7} (b + 1)^{7} (c + 1)^{7} > 7^{7} a^{4} b^{4} c^{4}$

MathematicsMaxima and MinimaJEE Advanced 1982Moderate

View in:English Hindi

If $a x^{2} + \frac{b}{x} \geq c$ for all positive $x$ where $a > 0$ and $b > 0$ show that $27 a b^{2} \geq 4 c^{3}$ .

MathematicsProperties of TrianglesJEE Advanced 2001Moderate

View in:English Hindi

If $Δ$ is the area of a triangle with side lengths $a, b, c$ , then show that $Δ \leq \frac{1}{4} (a + b + c) ab c$ . Also show that the equality occurs in the above inequality if and only if $a = b = c$ .

MathematicsLocation of RootsJEE Advanced 1995Easy

View in:English Hindi

Let $a, b, c$ be real. If $a x^{2} + b x + c = 0$ has two real roots $α$ and $β$ , where $α < - 1$ and $β > 1$ , then show that $1 + c / a + ∣ b / a ∣ < 0$ .

MathematicsLinear InequalitiesJEE Main 2002Easy

View in:English Hindi

If $a, b, c$ are distinct +ve real numbers and $a^{2} + b^{2} + c^{2} = 1$ then $ab + b c + c a$ is

(A)

less than 1

(B)

equal to 1

(C)

greater than 1

(D)

any real no.

Answer:A

MathematicsMonotonicityJEE Advanced 1990Moderate

View in:English Hindi

Show that $2 sin x + tan x \geq 3 x$ where $0 \leq x < π /2$ .

MathematicsBinomial Expansion for Positive Integral IndexJEE Advanced 1990Moderate

View in:English Hindi

Prove that $\frac{n ^{7}}{7} + \frac{n ^{5}}{5} + \frac{2 n ^{3}}{3} - \frac{n}{105}$ is an integer for every positive integer $n$ .

MathematicsConditional IdentitiesJEE Advanced 2000Easy

View in:English Hindi

In any triangle $A B C$ , prove that $cot \frac{A}{2} + cot \frac{B}{2} + cot \frac{C}{2} = cot \frac{A}{2} cot \frac{B}{2} cot \frac{C}{2}$ .

MathematicsSolution of System of Linear Equations (Matrix Method and Cramer's Rule)JEE Main 2008Easy

View in:English Hindi

Let $a, b, c$ be any real numbers. Suppose that there are real numbers $x, y, z$ not all zero such that $x = cy + b z, y = a z + c x$ , and $z = b x + a y$ . Then $a^{2} + b^{2} + c^{2} + 2 ab c$ is equal to

(A)

2

(B)

- 1

(C)

0

(D)

1

Answer:D

MathematicsMonotonicityJEE Advanced 1983Moderate

View in:English Hindi