Question

If $xy\left( {y - x} \right) = 2{a^3}$, at what point does y have a minimum value.
A. $\,a,2a$
B. $\,a, - a$
C. $\,2a,2a$
D. None of these

Answer 1

Hint: To find the minimum value, first we will differentiate the given function and then put it equals to zero. Then, we will find the relation between $x$ and $y$ to get to the final answer. Remember that the final answer would be either in terms of a or a constant value.

Complete step by step answer:
In the above question, it is given that $xy\left( {y - x} \right) = 2{a^3}$. We can also write it as $x{y^2} - {x^2}y = 2{a^3}$. Now we will differentiate the above equation with respect to x,
${y^2} + 2y{y^1}x - 2xy - {x^2}{y^1} = 0$
$\Rightarrow {y^2} - 2xy + {y^1}(2yx - {x^2}) = 0$

On transposing, we get
$ \Rightarrow {y^1} = \dfrac{{2xy - {y^2}}}{{2xy - {x^2}}}$
Now, to find the minimum value we will put it equals to zero.As we know that for a defined function, the denominator can not be equal to zero.Therefore,
$ \Rightarrow 2xy - {y^2} = 0$
$ \Rightarrow y\left( {2x - y} \right) = 0$

Now, there can be two cases either $y = 0$ or $2x - y = 0$. But y can not be equals to zero because it is not satisfying the equation which is given in the question.Therefore,
$2x - y = 0$
$ \Rightarrow y = 2x$
We have, $xy\left( {y - x} \right) = 2{a^3}$
Now, substitute $y = 2x$
$ \Rightarrow x\left( {2x} \right)\left( {2x - x} \right) = 2{a^3}$
$ \Rightarrow 2{x^3} = 2{a^3}$

On dividing both side by $2$, we get
$ \Rightarrow {x^3} = {a^3}$
Taking cube root both sides,
$ \Rightarrow x = a$
Now substitute the above value of x in $y = 2x$.
We get,
$ \Rightarrow y = 2a$
Therefore, the required point is $\left( {a,2a} \right)$.

Hence, the correct option is A.

Note: In this question, after differentiation when we put the value equals to zero, we get only a point, which is the point of minima. Therefore, we can say that this function has no point of maxima and there is only one point of minima. We can also cross check our answer by putting the given point in the equation.