Application of DerivativeHard
Question
If a1, a2 ........ an ∈ R+ and a1, a2 .......an = 1 then the least value of (1 + a1 + a21 ) (1 + a2 + a22 )....(1 + an + an2 ) is -
Options
A.3n
B.n3n
C.33n
D.data inadequate
Solution
Using AM ≥ GM,
≥ (1. a1a21)1/3 ⇒ 1 + a1 + a12 ≥ 3a1
⇒ 1 + a2 + a22 ≥ 3a2 ⇒ ....... 1 + an + a2n ≥ 3an
Multiplying these,
(1 + a + a21) .... (1 + an + a2n) ≥ 3n (a1a2a1 ... an) = 3n .1
≥ (1. a1a21)1/3 ⇒ 1 + a1 + a12 ≥ 3a1⇒ 1 + a2 + a22 ≥ 3a2 ⇒ ....... 1 + an + a2n ≥ 3an
Multiplying these,
(1 + a + a21) .... (1 + an + a2n) ≥ 3n (a1a2a1 ... an) = 3n .1
Create a free account to view solution
View Solution FreeMore Application of Derivative Questions
The tangent to the curve (x − 2)4 + ( y − 1)4 = 81 at the point (5, 1) is-...The point on the curve y2 = 8x for which the abscissa and ordinate change at the same rate is -...If a, b, c be non-zero real numbers such that (ax2 + bx + c)dx = (ax2 + bx + c) dx = 0, then the equation ax2 + bx + c =...The surface area of a cube is increasing at the rate of 2 cm2/sec. When its edge is 90 cm, the volume is increasing at t...At what point on the curve y = e−x , the tangent cuts intercept equal in length on coordinate axes-...