Question

The function \( f(x) = x^3 - 3x^2 + 12x - 18 \) is:

• A. strictly decreasing on \( \mathbb{R} \)
• B. strictly increasing on \( \mathbb{R} \)
• C. neither strictly increasing nor strictly decreasing on \( \mathbb{R} \)
• D. strictly decreasing on \( (-\infty, 0) \)

Hint:

A function is strictly increasing if its derivative is positive for all values of \( x \). Always check the sign of the derivative over the domain.

Answer 1

The Correct Option is B

Step 1: Find the derivative of \( f(x) \).
The derivative of \( f(x) \) is: \[ f'(x) = 3x^2 - 6x + 12. \] Step 2: Analyze \( f'(x) \).
Simplify \( f'(x) \): \[ f'(x) = 3(x^2 - 2x + 4). \] The quadratic \( x^2 - 2x + 4 \) has a discriminant: \[ \Delta = (-2)^2 - 4(1)(4) = 4 - 16 = -12. \] Since the discriminant is negative, \( x^2 - 2x + 4 \) is always positive. Hence, \( f'(x)>0 \) for all \( x \in \mathbb{R} \). Step 3: Conclusion about monotonicity.
Since \( f'(x)>0 \) for all \( x \), the function \( f(x) \) is strictly increasing on \( \mathbb{R} \). Step 4: Final Answer.
The function \( f(x) \) is: \[ \boxed{\text{strictly increasing on } \mathbb{R}}. \]