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:

To determine if a function is strictly increasing, examine the sign of its derivative. If the derivative is positive throughout the domain, the function is strictly increasing.

Answer 1

The Correct Option is B

Step 1: Calculate the derivative of \( f(x) \).
The derivative of the given function \( f(x) \) is: \[ f'(x) = 3x^2 - 6x + 12. \] Step 2: Simplify the derivative expression.
We can factor out the common term to simplify \( f'(x) \): \[ f'(x) = 3(x^2 - 2x + 4). \] The quadratic expression \( x^2 - 2x + 4 \) has a discriminant: \[ \Delta = (-2)^2 - 4(1)(4) = 4 - 16 = -12. \] Since the discriminant is negative, the quadratic expression is always positive. Thus, \( f'(x)>0 \) for all real values of \( x \). Step 3: Monotonicity Conclusion.
Given that \( f'(x)>0 \) for all \( x \in \mathbb{R} \), the function \( f(x) \) is strictly increasing on the entire real number line. Step 4: Final Conclusion.
Therefore, the function \( f(x) \) is: \[ \boxed{\text{strictly increasing on } \mathbb{R}}. \]