Find the intervals in which the function f given by f (x) = x³ + 1/x³, x ≠ 0 is
(i) Increasing   (ii) Decreasing

Solution:

Increasing functions are those functions that increase monotonically within a particular domain, and decreasing functions are those which decrease monotonically within a particular domain

We have f (x) = x³ + 1/x³

On differentiating wrt x, we get

f' (x) = 3x² - 3/x⁴

= (3x⁶ - 3)/x⁴

Now,

f' (x) = 0

3x⁶ - 3 = 0

x⁶ = 1

⇒ x = ± 1

Now, the points x = 1 and x = - 1 divide the real line into three disjoint intervals i.e., (- ∞, - 1) , (- 1, 1) and (1, ∞)

In intervals (- ∞, - 1) and (1, ∞) i.e., when x < - 1 and x > 1, f' (x) > 0

Thus, when x < - 1 and x > 1, f is increasing.

In interval (- 1, 1) i.e., when - 1 < x < 1, f' (x) < 0.

Thus, when - 1 < x < 1, f is decreasing

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NCERT Solutions Class 12 Maths - Chapter 6 Exercise ME Question 7

Find the intervals in which the function f given by f (x) = x³ + 1/x³, x ≠ 0 is (i) Increasing (ii) Decreasing

Summary:

The intervals in which the function f given by f (x) = x³ + 1/x³, x ≠ 0 is increasing when x < - 1 and x > 1, and decreasing when - 1 < x < 1