It's logical

Algebra Level 4

If three positive real numbers $a,b,c$ are in Arithmetic progression such that $a\times b \times c=4,$ then find the minimum value of $b$ .

The answer is 1.58740.

3 solutions

Bala Vidyadharan
Nov 18, 2015

We are given that $a, b,$ and $c$ are in arithmetic progression , so it must be that $2b=a+c.$ Now, we apply AM-GM to $a,b,$ and $c$ to get $\frac { a+b+c }{ 3 } \ge \sqrt [ 3 ]{ abc } \longrightarrow \frac { b+2b }{ 3 } \ge \sqrt [ 3 ]{ 4 } \longrightarrow b \ge 1.58$

So, the minimum value of $b$ is 1.58.

Generally for positive numbers, minima is attained when numbers are equal,
So, $a^3 = 4 \Rightarrow a=b=c=1.58$

Akhil Bansal - 5 years, 6 months ago

same way sir

PSN murthy - 5 years, 6 months ago

A.M>G.M. can be used to get the required properties

Akshay Sharma - 5 years, 6 months ago

Same way!!

Dev Sharma - 5 years, 6 months ago

Me too!!!!

Kushagra Sahni - 5 years, 6 months ago

Hey guys can you prove a+b+c>4 For this

Ravinaka Hewa - 5 years, 6 months ago

Great......

Dev Sharma - 5 years, 6 months ago

Vinod Kumar
Jun 26, 2019

Find x^3=4 with x=1.5874 as the solution

Akshay Sharma
Nov 26, 2015

Let a=x-d , b=x , c=x+d Now using A.M>G.M in three terms a,b,c; we get x>4^(0.33334) x>~1.587

It's logical

The answer is 1.58740.

3 solutions

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