Question

The product of two positive numbers is 1024. what is the minimum value of their sum?

Answer 1

Let the two positive numbers are x , y and there sum is s
So,
            x y = 1024 ⇒⇒⇒⇒ (1)
And        S = x + y ⇒⇒⇒⇒ (2)
by substituting from (1) at (2) with the value of y = 1024/x

∴ s = x + $\frac{1024}{x}$

Differentiating both sides with respect to x to find the minimum value of the sum and equating to zero

∴  $\frac{ds}{dx} = 1 - \frac{1024}{ x^{2} } = 0$

solve the last equation for x
∴ $1 - \frac{1024}{ x^{2} } = 0$
    $1 = \frac{1024}{ x^{2} }$  ⇒⇒⇒ *x²
∴ x²  = 1024
∴ $x = \pm \sqrt{1024} = \pm 32$
taking the positive number as stated in the problem

x = 32

So, The numbers are 32 , 32
And theire mimimum sum = 32 + 32 = 64