Two positive numbers have a sum of 3636. What is the maximum product of one number times the...

Question:

Two positive numbers have a sum of 3636. What is the maximum product of one number times the square of the second number?

Maximum Value:

The maximum and minimum value of a function can be calculated using the first and second derivative of the function. For the critical points, we use the first derivative, and for the maximum and minimum value, we use the second derivative.

Answer and Explanation:


Let us consider p and q are the two positive numbers.

According to the statement in the problem the sum of two positive numbers is equal to:

{eq}\begin{align*} p + q &= 3636\\ p &= 3636 - q \end{align*} {/eq}

The maximum product of one number time the square of the second number is:

{eq}x = p{q^2}\cdots\cdots\rm{(1)} {/eq}


Substitute p into the above equation.

{eq}\begin{align*} x &= \left( {3636 - q} \right){q^2}\\ x &= 3636{q^2} - {q^3} \end{align*} {/eq}


We will differentiate the above equation to obtain the critical points.

{eq}\begin{align*} x' &= 7272q - 3{q^2}\\ &= q\left( {7272 - 3q} \right) \end{align*} {/eq}


For critical points the first derivative must be equal to zero as:

{eq}\begin{align*} x' = 0\\ q\left( {7272 - 3q} \right) &= 0\\ q &= 0,\;2424 \end{align*} {/eq}


Now, we will find the second derivative to obtain the maximum and minimum value.

{eq}x'' = 7272 - 6q {/eq}


By using critical points in the above equation, we will get maximum and minimum value:

{eq}\begin{align*} x'' &= 7272 - 6\left( 0 \right)\\ &= 7272 \end{align*} {/eq}

Also,

{eq}\begin{align*} x'' &= 7272 - 6\left( {2424} \right)\\ &= - 7272 \end{align*} {/eq}


According to second derivative, at q=2424 the value of second derivative is negative hence it will be maximum.


The value of the number p is calculated as,

{eq}\begin{align*} p &= 3636 - 2424\\ &= 1212 \end{align*} {/eq}


Finally, we will substitute the value of the numbers in equation 1.

{eq}\begin{align*} x &= \left( {1212} \right){\left( {2424} \right)^2}\\ &= 7121440512 \end{align*} {/eq}


Thus, the maximum product of one number time the square of the second number is 7121440512


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Finding Minima & Maxima: Problems & Explanation

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