Question

Halle el menor valor posible de la expresion (x + 1 /x ) (x + 81 /x );

Answer 1

Para saber el valor mínimo, hay que derivar la función tal que

f'(x) = 0

g(x) = x + 1/x

g'(x) = 1 - 1/x²

h(x) = x + 81/x

h'(x) = 1 - 81/x²

f(x) = g(x)•h(x)

f'(x) = g'(x)•h(x) + g(x)•h'(x)

f'(x) = (1 - 1/x²)•(x + 81/x) + (x + 1/x)•(1 - 81/x²)

f'(x) = x + 81/x - 1/x - 81/x³ + x - 81/x + 1/x - 81/x³

f'(x) = 2(x - 81/x³)

0 = 2(x - 81/x³)

0 = x - 81/x³

x = 81/x³

x⁴ = 81

|x| = 3

(3 + 1/3)•(3 + 81/3) = (10/3)•(30) = 100

(-3 + 1/(-3))•(-3 + 81/(-3)) = (-10/3)•(-30) = 100

Rta: su menor valor es 100