# Prove that the function given by f (x) = x^{3} - 3x^{2} + 3x - 100 is increasing in R

**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) = x^{3} - 3x^{2} + 3x - 100

Therefore,

On differentiating wrt x, we get

f' (x) = 3x^{2} - 6x + 3

Changing into whole square form

= 3(x^{2} - 2x + 1)

= 3(x - 1)^{2}

For x ∈ R ,

(x - 1)^{2} ≠ 0

So, f' (x) is always positive in R.

If f' (x) > 0 then the function is strictly increasing.

Thus,

the function is increasing in R

NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.2 Question 18

## Prove that the function given by f (x) = x^{3} - 3x^{2} + 3x - 100 is increasing in R

**Summary:**

Hence we have concluded that the function given by f (x) = x^{3} - 3x^{2} + 3x - 100 is increasing in R. If f' (x) > 0 then the function is strictly increasing