The sum of two positive numbers is 10. What two numbers will maximize the product?

Question:

The sum of two positive numbers is {eq}10 {/eq}. What two numbers will maximize the product?

Finding Minima & Maxima:

When we have the sum of the number given, then by differentiation method and the simplification technique we can easily prove that the product will be maximum on the condition when the two numbers are the same.

Let the two numbers be:

{eq}x {/eq} and {eq}10-x {/eq}

Their product is given by:

$$p=x(10-x)$$

Now to maximize the product function, we will differentiate it to zero and solve for {eq}x{/eq}:

\begin{align} \frac{dp}{dx} &=0\\[0.2cm] \frac{d}{dx}\left(x\left(10-x\right)\right)&=0\\[0.2cm] 10-2x&=0\\[0.2cm] x&=5 \end{align}

So both numbers should be equal to {eq}5{/eq} to get the maximum product.