Question

Show that the function `f(x) = x/3 + 3/x` decreases in the intervals (–3, 0) ∪ (0, 3).

Answer 1

`f(x) = x/3 + 3/x`

f'(x) = `1/3 - 3/x^2`

f'(x) = 0

`1/3 - 3/x^2 = 0`

`(x^2 - 9)/(3x^2) = 0`

x² – 9 = 0, x not equal sig 0

(x + 3)(x – 3) = 0

x = –3, 3

Here x ≠ 0 the function decreases in the interval (–3, 0) ∪ (0, 3).