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BITSAT 2024

Exam: BITSAT
Year: 2024
Subject: Mathematics
Chapter: Sequences and Series

BITSAT 2024 · Mathematics · Sequences and Series

If a>0, b>0, c>0 and a, b, c are distinct, then (a+b)(b+c)(c+a) is...

If

a>0, b>0, c>0

and

a, b, c

are distinct, then

(a+b)(b+c)(c+a)

is greater than

Options

A $2(a+b+c)$
B $3(a+b+c)$
C $6 a b c$
D $8 a b c$

Correct answer

D. $8 a b c$

Step-by-step solution

We know that

\mathrm{AM} \geq \mathrm{GM}

...(i) Now, applying Eq. (i) in

a

and

b\Rightarrow \quad \frac{a+b}{2} \geq \sqrt{a b}

....(ii) applying Eq. (i) in

b

and

c\Rightarrow \quad \frac{b+c}{2} \geq \sqrt{b c}

....(iii) and applying Eq. (i) in

c

and

a\Rightarrow \quad \frac{c+a}{2} \geq \sqrt{c a}

....(iv) By E'qs. (ii), (iii) and (iv)

\begin{array}{cc} \Rightarrow & \frac{(a+b)(b+c)(c+a)}{8} \geq a b c \\ \Rightarrow & (a+b)(b+c)(c+a) \geq 8 a b c \end{array}

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