Suppose f(x) is a quadratic function. The slope of f(x) at the extremum point is a.0b.infinitec.negatived.positive

Analyzing the Quadratic Function

1. Understanding the Nature of a Quadratic Function
   A quadratic function can generally be represented as $f(x) = ax^2 + bx + c$. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the coefficient $a$.

2. Identifying the Extremum Point
   The extremum point (maximum or minimum) of the quadratic function occurs at the vertex of the parabola. The x-coordinate of the vertex can be found using the formula $x = -\frac{b}{2a}$.

3. Slope at the Extremum Point
   The slope of a function at any point is given by the derivative of that function. The derivative of the quadratic function $f(x)$ is $f'(x) = 2ax + b$. Evaluating this at the vertex, we substitute $x = -\frac{b}{2a}$ into the derivative:

   $f'\left(-\frac{b}{2a}\right) = 2a\left(-\frac{b}{2a}\right) + b = -b + b = 0$

Final Answer

The slope of $f(x)$ at the extremum point is 0 (Option a).