Answer

Verified

356.1k+ views

**Hint**: In the above question it is given that ${x^2}{y^3}$. So on the basis of this we will break down the term of $x$ in twice as there is ${x^2}$. So similarly we will break the term of $y$ in three parts as it is given that ${y^3}$. Now we have to find the minimum value of $3x + 4y$. So we will use the property of arithmetic mean and geometric mean to solve this question.

**:**

__Complete step-by-step answer__We will break the term $3x$ into $\dfrac{{3x}}{2},\dfrac{{3x}}{2}$ and we will break the another term $4y = \dfrac{{4y}}{3},\dfrac{{4y}}{3},\dfrac{{4y}}{3}$.

Now we will use the property of arithmetic mean (AM) and geometric mean (GM) . It says that $AM \geqslant GM$.

We know that formula of arithmetic mean is i.e. $\overline X = \dfrac{{\sum\limits_{i = 1}^n {{X_i}} }}{N}$, where $N$ is the total number of observations, here we have $N = 5$. Now applying the formula we solve it.

So we have AM of the above terms is

$AM = \dfrac{{\dfrac{{3x}}{2} + \dfrac{{3x}}{2} + \dfrac{{4y}}{3} + \dfrac{{4y}}{3} + \dfrac{{4y}}{3}}}{5}$.

We have to add all these terms so, by taking the LCM of the denominator and adding we have the value

$AM = \dfrac{{\dfrac{{9x + 9x + 8y + 8y + 8y}}{6}}}{5} = \dfrac{{\dfrac{{6(3x + 4y)}}{6}}}{5}$.

We have taken the common factor out in the numerator and then it gets cancelled with the denominator, so it gives us $\dfrac{{3x + 4y}}{5}$.

Now we will calculate the value of

GM$ = {\left( {\dfrac{3}{2} \times \dfrac{3}{2} \times \dfrac{4}{3} \times \dfrac{4}{3} \times \dfrac{4}{3}} \right)^{\dfrac{1}{5}}}$. On further solving we have ${(2 \times 2 \times 2 \times 2 \times 2)^{\dfrac{1}{5}}} = {2^{5 \times \dfrac{1}{5}}}$.

It gives us the value $GM = 2$.

Now by putting the values back in the formula we have: $\dfrac{{3x + 4y}}{5} \geqslant 2$, on solving we have $3x + 4y \geqslant 2 \times 5 \Rightarrow 3x + 4y \geqslant 10$. Here we can see that the value of $3x + 4y$ has to be greater or equal to $10$, it cannot be less than it.

Hence we can say that the minimum value of $3x + 4y$ is $10$.

**So, the correct answer is “10”.**

**Note**: We should know that if in AM, there is set of numbers ${a_1},{a_2},{a_3}...{a_n}$, then the value of AM is $\left( {\dfrac{{{a_1} + {a_2} + {a_3} + ...{a_n}}}{n}} \right)$, where $n$ is the number of terms. Similarly in G.M if set of values as ${a_1},{a_2},{a_3}...{a_n}$, then the $GM = {\left( {{a_1} \times {a_2} \times {a_3}...{a_n}} \right)^{\dfrac{1}{n}}}$, where $n$ is the number of terms. There is an inequality relation between the AM and GM , greater than equal to i.e. $AM \geqslant GM$.

Recently Updated Pages

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Advantages and disadvantages of science

10 examples of friction in our daily life

Trending doubts

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Which are the Top 10 Largest Countries of the World?

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

10 examples of law on inertia in our daily life

Write a letter to the principal requesting him to grant class 10 english CBSE

In 1946 the Interim Government was formed under a Sardar class 11 sst CBSE

Change the following sentences into negative and interrogative class 10 english CBSE