The product of two positive numbers is 1024. what is the minimum value of their sum?
Accepted Solution
A:
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 + [tex] \frac{1024}{x} [/tex]
Differentiating both sides with respect to x to find the minimum value of the sum and equating to zero
solve the last equation for x ∴ [tex]1 - \frac{1024}{ x^{2} } = 0[/tex] [tex] 1 = \frac{1024}{ x^{2} } [/tex] ⇒⇒⇒ *x² ∴ x² = 1024 ∴ [tex] x = \pm \sqrt{1024} = \pm 32[/tex] taking the positive number as stated in the problem
x = 32
So, The numbers are 32 , 32 And theire mimimum sum = 32 + 32 = 64