# Which of the following is true regarding the function f(x) = x3 - 6x2 + 12x - 18 ?

51 views
in Calculus
closed
Which of the following is true regarding the function f(x) = x3 - 6x2 + 12x - 18 ?
1. f(x) is an increasing function on R
2. f(x) is a decreasing function on R
3. f(x) is neither an increasing function on R nor a decreasing function on R.
4. None of these
5. Both 1 and 2

by (30.1k points)
selected

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.