Find the values of x for which y = [x (x - 2)]² is an increasing function

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,

y = [x (x - 2)]²

= [x² - 2x]²

Therefore,

dy/dx = d/dx [x² - 2x]²

= 2(x² - 2x)(2x - 2)

= 4x (x - 2)(x - 1)

Now, dy/dx = 0

Hence,

⇒ 4x (x - 2)(x - 1)

⇒ x = 0, x = 2, x = 1

x = 0, x = 1 and x = 2 divide the number line intervals (- ∞, 0), (0, 1), (1, 2) and (2, ∞)

In (- ∞, 0) and (1, 2),

dy/dx < 0

Hence, y is strictly decreasing in intervals (- ∞, 0) and (1, 2)

In (0, 1) and (2, ∞),

dy/dx > 0

Hence, y is strictly increasing in intervals (0, 1) and (2, ∞)

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NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.2 Question 8

Find the values of x for which y = [x (x - 2)]² is an increasing function

Summary:

The  values of x for which y = [x (x - 2)]² is an increasing function is in the intervals (0, 1) and (2, ∞). x = 0, x = 1 and x = 2 divide the number line intervals (- ∞, 0), (0, 1), (1, 2) and (2, ∞)