Find the intervals in which the function f given by f(x) is

(i) Increasing

(ii) Decreasing

⇒ y = x³ + x^-3

Now to find the intervals in which f(x) increases and decreases we first have to find the critical points

Critical points are those points or values of x at which f’(x) = 0

Hence differentiate y with respect to x and equate it to 0

⇒ x⁶ – 1 = 0

⇒ x⁶ = 1

By observation x satisfies both 1 and -1 hence

⇒ x = -1 and x = 1

Now we have to check for intervals (-∞, -1), (-1, 1) and (1, ∞)

Case1:

When x < -1 that is x in (-∞, -1)

Observe that x⁶ is positive and greater than 1 so (x⁶ – 1) is also positive and the denominator x⁴ is also positive because of even power

Hence hence f(x) is increasing

Case 2:

When -1 < x < 1 that is x in (-1, 1)

Observe that x⁶ is positive but less than 1 so (x⁶ – 1) is negative and the denominator x⁴ is positive because of even power hence is negative because of (x⁶ – 1)

Hence hence f(x) is decreasing

Case 3:

When x > 1 that is x in (1, ∞)

Observe that x⁶ is positive and greater that 1 so (x⁶ – 1) is positive and the denominator x⁴ is positive because of even power

Hence hence f(x) is increasing

Hence the conclusion of case1, case2 and case3 is that f(x) is increasing in the interval (-∞, -1) U (1, ∞) and decreasing in the interval (-1, 1)