Step-by-step Solution:

To determine the conditions under which the function \( f(x) = x^3 + ax^2 + bx + c \) has no extremum, we can proceed with the following steps: 1. Find the First Derivative: \[ f'(x) = \frac{d}{dx} (x^3 + ax^2 + bx + c) = 3x^2 + 2ax + b \] 2. Condition for No Extremum: For the function to have no extremum, the first derivative must be positive for all \( x \): \[ 3x^2 + 2ax + b > 0 \] 3. Quadratic Inequality: The quadratic expression \( 3x^2 + 2ax + b \) will be positive for all \( x \) if: - The coefficient of \( x^2 \) is positive (\( 3 > 0 \)), which is always true. - The discriminant \( D \) is negative: \[ D = (2a)^2 - 4 \times 3 \times b < 0 \] \[ 4a^2 - 12b < 0 \] \[ a^2 - 3b < 0 \] \[ a^2 < 3b \] Therefore, the condition for the function \( f(x) = x^3 + ax^2 + bx + c \) to have no extremum is: \[ \boxed{a^2 < 3b} \]