Question

a, b, c are three positive numbers and $abc^2$ has the greatest value $\dfrac{1}{{64}}$. Then
A.\[a = b = \dfrac{1}{2},c = \dfrac{1}{4}\]
B.\[a = b = \dfrac{1}{4},c = \dfrac{1}{2}\]
C.\[a = b = c = \dfrac{1}{3}\]
D.None of these

Answer 1

Hint: In this question, we need to establish a relation between the terms a, b and c such that $abc^2$ has the greatest value $\dfrac{1}{{64}}$. For this we will use the inequalities properties between Am and GM which is given as \[\dfrac{{{x_1} + {x_2} + ..... + {x_n}}}{n} \geqslant \sqrt[n]{{{x_1}.{x_2}......{x_n}}}\].

Complete step-by-step answer:
a, b, c are three positive number
We know the inequality of three number in AM and GM is given as
\[\dfrac{{{x_1} + {x_2} + {x_3}}}{3} \geqslant \sqrt[3]{{{x_1}{x_2}{x_3}}} - - (i)\]
Hence by using equation (i) we can write
\[\dfrac{{a + b + {c^2}}}{3} \geqslant \sqrt[3]{{ab{c^2}}} - - (ii)\]
Where the maximum value of
\[ab{c^2} = \dfrac{1}{{64}}\]
Since we are asked to find the value of a, b and c, hence we will use the equation (iii), where the equation will be equal if the value of\[a = b = {c^2} - - (iii)\],
By checking the options
\[a = b = \dfrac{1}{2},c = \dfrac{1}{4}\]
In equation (iii), if we assume
\[a = b = {c^2} = \dfrac{1}{2}\]
Hence the values of\[c = \dfrac{1}{{\sqrt 2 }}\],
But in the option value of \[c = \dfrac{1}{4}\]
Therefore we can say given values are incorrect.
\[a = b = \dfrac{1}{4},c = \dfrac{1}{2}\]
In equation (iii), if we assume
\[a = b = {c^2} = \dfrac{1}{4}\]
Hence the values of\[c = \dfrac{1}{2}\],
Also in the option value of \[c = \dfrac{1}{2}\]
Therefore we can say given values are correct.
\[a = b = c = \dfrac{1}{3}\]
In equation (iii), if we assume
\[a = b = {c^2} = \dfrac{1}{3}\]
Hence the values of\[c = \dfrac{1}{{\sqrt 3 }}\],
Also in the option value of \[c = \dfrac{1}{3}\]
Therefore we can say given values are incorrect.
So, the correct answer is “Option B”.

Note: In AM and GM the inequality of n positive numbers is given as
\[\dfrac{{{x_1} + {x_2} + ..... + {x_n}}}{n} \geqslant \sqrt[n]{{{x_1}.{x_2}......{x_n}}}\]
For a set of positive numbers\[{a_1},{a_2},.......,{a_n}\], the arithmetic mean is given as \[AM = \dfrac{{{a_1} + {a_2} + ........ + {a_n}}}{n}\]
Also for a set of positive numbers\[{a_1},{a_2},.......,{a_n}\], their geometric mean is given as
\[GM = \sqrt {{a_1},{a_2},........,{a_n}} \]