Question

Prove that the function f(x) = x³ – 6x² + 12x – 18 is increasing on R.

Answer 1

Given:- Function f(x) = x³ – 6x² + 12x – 18

Theorem:- Let f be a differentiable real function defined on an open interval (a, b).

(i) If f’(x) > 0 for all x ∈ (a, b), then f(x) is increasing on (a, b)

(ii) If f’(x) < 0 for all x ∈ (a, b), then f(x) is decreasing on (a, b)

Algorithm:-

(i) Obtain the function and put it equal to f(x)

(ii) Find f’(x)

(iii) Put f’(x) > 0 and solve this inequation.

For the value of x obtained in (ii) f(x) is increasing and for remaining points in its domain, it is decreasing.

Here we have,

Hence, condition for f(x) to be increasing

Thus f(x) is increasing on interval x ∈ R