Home
Class 12
MATHS
If a,b,c and d are four positive real nu...

If a,b,c and d are four positive real numbers such that abcd=1 , what is the minimum value of `(1+a)(1+b)(1+c)(1+d)`.

A

1

B

4

C

16

D

64

Text Solution

Verified by Experts

The correct Answer is:
C
`:.(1+a)(1+b)(1+c)(1+d)` ,brgt `=1+a+b+c+d+ab+ac+ad+bc+bd+cd+abc+abd+cda+cdb+abcd " " [" 16 terms "]`
`:.AMgeGM`
`((1+a)(1+b)(1+c)(1+d))/(16)ge (a^(8)b^(8)c^(8)d^(8))^((1)/(16))`
`=(abcd)^((1)/(2))=(1)^((1)/(2))=1 " " [:. Abcd=1]`
`((1+a)(1+b)(1+c)(1+d))/(16)ge1`
`(1+a)(1+b)(1+c)(1+d)ge16`
`:.` Minimum value of `(1+a)(1+b)(1+c)(1+d)` is 16.
Promotional Banner

Similar Questions

Explore conceptually related problems

If three positive real numbers a,b,c are in AP such that abc=4, then the minimum value of b is

If a, b, c are three positive real numbers such that abc^(2) has the greatest value (1)/(64) , then

If a,b,c and d are odd natural numbers such that a+b+c+d=20, the number of values of the ordered quadruplet (a,b,c,d) is

If a,b,c and d are four real numbers of the same sign, then the value of (a)/(b)+(b)/(c )+(c )/(d)+(d)/(a) lies in the interval

If a,b,c,d are positive real numbers such that (a)/(3) = (a+b)/(4)= (a+b+c)/(5) = (a+b+c+d)/(6) , then (a)/(b+2c+3d) is:-

If a,b and c are three consecutive positive integers such that 1/(a!)+1/(b!)=lambda/(c!) then the value of lambda ;

If a,b,c are non-zero real numbers, then the minimum value of the expression ((a^(8)+4a^(4)+1)(b^(4)+3b^(2)+1)(c^(2)+2c+2))/(a^(4)b^(2)) equals

If (log)_b a(log)_c a+(log)_a b(log)_c b+(log)_a c(log)_bc=3 (where a , b , c are different positive real numbers !=1), then find the value of a b c .

Suppose a, b, c are positive integers with altbltc such that 1//a+1//b+1//c=1 . The value of (a+b+c-5) is …………

Let a,b,c,d be positive real numbers with altbltcltd . Given that a,b,c,d are the first four terms of an AP and a,b,d are in GP. The value of (ad)/(bc) is (p)/(q) , where p and q are prime numbers, then the value of q is