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

Concept: A function is increasing for all real $x$ if: \[ f'(x) \ge 0 \; \forall x \]

Step 1: {Differentiate function.}
\[ f'(x)=3x^{2}-k \]

Step 2: {For increasing function, require:}
\[ 3x^{2}-k \ge 0 \]

Step 3: {Minimum value of $3x^{2}$ is 0.}

Step 4: {Thus condition becomes:}
\[ -k \ge 0 \]

Step 5: {Final result:}
\[ k \ge 0 \]