Correct Answer - Option 1 : f(x) is an increasing function on R
Concept:
Let f(x) be a function defined on an interval (a, b), this function is said to be an increasing function:
- If x1 < x2 then f(x1) ≤ f(x2) ∀ x1, x2 ∈ (a, b)
- Here, \(\frac{{dy}}{{dx}} \ge 0\;or\;f'\left( x \right) \ge 0\)
Similarly, f(x) is said to be a decreasing function:
- If If x1 < x2 then f(x1) ≥ f(x2) ∀ x1, x2 ∈ (a, b).
- Here, \(\frac{{dy}}{{dx}} \le 0\;or\;f'\left( x \right) \le 0\)
Calculation:
Given: f(x) = x3 - 6x2 + 12x - 18
Let's find out f'(x)
⇒ f'(x) = 3x2 - 12x + 12 = 3 ⋅ (x2 - 4x + 4)
⇒ f'(x) = 3 ⋅ (x - 2)2
Now as we know that, for any x ∈ R we have (x - 2)2 ≥ 0
⇒ f'(x) ≥ 0
As we know that for an increasing function say f(x) we have f'(x) ≥ 0
Hence, the given function f(x) is an increasing function on R.