Find the positive number such that the sum of 8 times this number and 7 times its reciprocal is as small as possible.

√56/8

Step-by-step explanation:

Let the number be x

f (x) = 8x + 7 (1/x)

f (x) = 8x + 7/x

Differentiate f (x) with respect to x

f' (x) = 8x - 7/x = 0

8 - 7/x^2 = 0

(8x^2 - 7) / 2 = 0

8x^2 - 7 = 0

8x^2 = 7

x^2 = 7/8

x = √7/8

x = √7 / √8

x = (√7/√8) (√8/√8)

x = (√7*√8) / √8*√8)

x = √56/8