Find a positive number x such that the sum of 25x and 1/x is as small as possible.

Step-by-step explanation:

Positive number of x = ?

Such that the sum of numbers 25x and 1/x is as small as possible.

Step 1:

Let's use differentiation to determine the minimum sum

Y = 25x + 1/xdy

Y = 25x + x^-1 dy

dy/dx = 25.x^(1-1) + x^(-1-1)

= 25.x^0 + x^(-2)

= 25(1) + x^-2

= 25 + 1/x^2

Step 2:

Let's find the value of x

Take dy/dX = 0

0 = 25 + 1/x^2

(Make x the subject of the formula)

25 = 1/x^2

1/25 = x^2

(find the square root)

1/5 = x

Step 3:

Substitute the value of x (that is our calculated 1/5) into the question so as to determine the sum of 25x and 1/x

Sum = 25x + 1/x

= 25.(1/5) + 1/(1/5)

= 5 + 5

= 10

Therefore,

When x = 1/5, the sum of 25x and 1/x is 10

That's the minimum value over the open interval x>0.

I hope this helps.