Let a_1, a_2, a_3 be any positive real number | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) We use the $AM \ge GM$ inequality for positive real numbers.For option $A$: By $AM \ge GM$,$\frac{a_1^3 + a_2^3 + a_3^3}{3} \ge \sqrt[3]{a_1^3 a_2

Vedclass · Accepted Answer

$(a_1 + a_2 + a_3) \left( \frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} \right)^3 \le 27$

Vedclass · Answer

(D) We use the $AM \ge GM$ inequality for positive real numbers.For option $A$: By $AM \ge GM$,$\frac{a_1^3 + a_2^3 + a_3^3}{3} \ge \sqrt[3]{a_1^3 a_2^3 a_3^3} = a_1 a_2 a_3$,so $a_1^3 + a_2^3 + a_3^3 \ge 3a_1 a_2 a_3$. This is true.For option $B$: By $AM \ge GM$,$\frac{a_1/a_2 + a_2/a_3 + a_3/a_1}{3} \ge \sqrt[3]{\frac{a_1}{a_2} \cdot \frac{a_2}{a_3} \cdot \frac{a_3}{a_1}} = 1$,so $\frac{a_1}{a_2} + \frac{a_2}{a_3} + \frac{a_3}{a_1} \ge 3$. This is true.For option $C$: By $AM \ge HM$,$(a_1 + a_2 + a_3) \ge \frac{9}{1/a_1 + 1/a_2 + 1/a_3}$,which implies $(a_1 + a_2 + a_3) \left( \frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} ight) \ge 9$. This is true.For option $D$: The inequality $(a_1 + a_2 + a_3) \left( \frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} ight)^3 \le 27$ is generally false. For example,if $a_1=a_2=a_3=1$,then $(3)(3^3) = 81 
ot\le 27$. Thus,option $D$ is not true.

Let $a_1, a_2, a_3$ be any positive real numbers,then which of the following statements is not true?

Similar Questions

If the $A.M.$ of two numbers is greater than the $G.M.$ of the numbers by $2$ and the ratio of the numbers is $4:1$,then the numbers are

If the arithmetic mean of two numbers is $A$ and the geometric mean is $G$,then the numbers are

If $a, b, c$ are in $A.P.;$ $b, c, d$ are in $G.P.$ and $\frac{1}{c}, \frac{1}{d}, \frac{1}{e}$ are in $A.P.,$ prove that $a, c, e$ are in $G.P.$

If the sum of the arithmetic mean and the harmonic mean between two positive numbers is $25$ and their geometric mean is $12$,then what is the sum of the numbers?

If $a, b, c$ are in $A.P.$ and $b, c, d$ are in $H.P.$,then

Vedclass Test Series

Exam Paper Generator

Online Exam Module