Question

If the function $f(x)=x^{3}-kx$ is increasing for all real x, then:

• A. $k \ge 0$
• B. $k \le 0$
• C. $k > 0$
• D. $k < 1$

Hint:

For polynomials, always analyze derivative minimum value to check monotonicity.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A function \( f(x) \) is increasing if \( f'(x) \ge 0 \) for all \( x \).

Step 2: Detailed Explanation:
\( f'(x) = 3x^2 - k \).
For \( f(x) \) to be increasing, \( 3x^2 - k \ge 0 \) for all \( x \).
This means \( 3x^2 \ge k \) for all \( x \).
Since \( 3x^2 \) is always \( \ge 0 \), this condition can only hold for all \( x \) if \( k \le 0 \).

Step 3: Final Answer:
\( k \le 0 \).