Sequences & SeriesmediumFree

Minimum of (1+a)(1+b)(1+c)(1+d) when abcd=1 | JEE

JEE Maths question with a full step-by-step solution.

Question
If a,b,c,da, b, c, d are positive reals with abcd=1abcd=1, then the minimum value of (1+a)(1+b)(1+c)(1+d)(1+a)(1+b)(1+c)(1+d) is
A44
B11
C1616correct
D1818
Solution
Step 1: AM–GM on each factor:
1+a≥2a,1+b≥2b,1+c≥2c,1+d≥2d.1+a\ge 2\sqrt a,\quad 1+b\ge 2\sqrt b,\quad 1+c\ge 2\sqrt c,\quad 1+d\ge 2\sqrt d.
 Step 2: Multiply:
(1+a)(1+b)(1+c)(1+d)≥16abcd=16.(1+a)(1+b)(1+c)(1+d)\ge 16\sqrt{abcd}=16.
 Step 3: Equality at a=b=c=d=1a=b=c=d=1, so the minimum is 16.16. Correct answer: (3)
Still stuck on this question?Ask your doubt on WhatsApp
Similar questions
Sequences & Series · easy
a, b, c, d are in increasing G.P. If the A.M. between a and b is 6 and the A.M. between c…
Sequences & Series · medium
Let a₁, a₂, …, a(4001) be in A.P. such that (1)/(a₁ a₂)+(1)/(a₂ a₃)+⋯+ 1 a(4000) a(4001)…
Sequences & Series · medium
If (p+q· 5^x)/(p-q· 5^x)=(q+r· 5^x)/(q-r· 5^x)=(r+s· 5^x)/(r-s· 5^x) , then p, q, r, s ar…
Sequences & Series · easy
Let a₁, a₂, … and b₁, b₂, … be arithmetic progressions with a₁=25 , b₁=75 and a(100)+b(10…

Solve more, learn faster

Sign up free to solve more JEE Maths questions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.

Sign up freeExplore doMath