Prove that the function f given by f (x) = x² - x + 1 is neither strictly increasing nor strictly decreasing on (- 1, 1)

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.

The given function is

f(x) = x² - x + 1

Therefore,

f' (x) = 2x - 1

Now,

f' (x) = 0

⇒ x = 1

x = 1/2 divides the interval into (- 1, 1/2) and (1/2, 1)

In interval (1/2, 1),

f' (x) = 2x - 1 > 0

Hence,

f is strictly decreasing in (- 1, 1/2)

In interval (1/2, 1),

f' (x) = 2x - 1 > 0

Hence, f is strictly increasing in (1/2, 1)

Thus, f is neither strictly increasing nor strictly decreasing in the interval (- 1, 1)

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NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.2 Question 11

Prove that the function f given by f (x) = x² - x + 1 is neither strictly increasing nor strictly decreasing on (- 1, 1).

Summary:

For the function: f (x) = x² - x + 1 is neither strictly increasing nor strictly decreasing in the interval (- 1, 1)