Explain how the first derivative test can be used to determine local minima and maxima for a function f(x).

The First Derivative Test is a method used to determine the local minima and maxima of a function \( f(x) \) by analyzing the sign changes of the first derivative of the function. Here's how the First Derivative Test works: 1. **Find the First Derivative**: Calculate the first derivative of the function \( f(x) \), denoted as \( f'(x) \). 2. **Find Critical Points**: Determine the critical points of the function by setting the first derivative equal to zero and solving for \( x \). These critical points are where the slope of the function changes. 3. **Test Intervals**: Divide the domain of the function into intervals based on the critical points. Test the sign of the first derivative in each interval. 4. **Determine Local Minima and Maxima**: - If the first derivative changes from positive to negative at a critical point, then that point is a local maximum. - If the first derivative changes from negative to positive at a critical point, then that point is a local minimum. By analyzing the sign changes of the first derivative around the critical points, we can determine the local minima and maxima of the function \( f(x) \).