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The first derivative test is a critical tool in calculus used to identify relative maxima and minima of a given function. It involves examining the sign changes of the first derivative \( f'(x) \) around critical points:

- **Step 1:** **Find the First Derivative**: This step requires determining the derivative of the function using appropriate differentiation techniques, such as the chain rule or product rule. For example, if you have \( f(x) = \sin^2 x \), its first derivative would be \( f'(x) = \sin 2x \).

- **Step 2:** **Locate Critical Points**: The critical points are determined where the first derivative equals zero or is undefined. For \( \sin 2x \), setting it to zero leads to solutions \( x = \frac{n\pi}{2} \) within a specified range, marking potential extrema.

- **Step 3:** **Apply the Test**: You must evaluate the sign of the derivative in intervals around each critical point. If the derivative changes from positive to negative at a critical point, it's a local maximum. Conversely, changing from negative to positive indicates a local minimum. For instance, at \( x = \frac{\pi}{2} \), if \( f'(x) > 0 \) on one side and \( < 0 \) on the other, you have a maximum.

The second derivative test enhances your analysis by confirming the nature of critical points. Here's how to apply it:

- **Step 1:** **Determine the Second Derivative**: Differentiate the first derivative to find the second derivative \( f''(x) \). If \( f'(x) = \sin 2x \), the second derivative would be \( f''(x) = 2 \cos 2x \).

- **Step 2:** **Evaluate at Critical Points**: Plug the critical points into the second derivative to ascertain the nature of each critical point.

• If \( f''(c) > 0 \), this indicates a local minimum at \( c \).
• If \( f''(c) < 0 \), there is a local maximum at \( c \).
• If \( f''(c) = 0 \), the test is inconclusive, and further investigation is needed.

For example, at \( x = \frac{\pi}{2} \), \( f''(\frac{\pi}{2}) = -2 \) suggests a local maximum, confirming observations from the first derivative test.

Critical points are essential for understanding a function's behavior and occur where the derivative equals zero or is undefined. To identify these points, follow these steps:

- **Step 1:** **Calculate First Derivative**: Begin by differentiating the function to find \( f'(x) \).
- **Step 2:** **Solve for Zeroes or Undefined Values**: Setting \( f'(x) = 0 \) helps locate potential critical points. In some cases, the derivative might be undefined at certain points, which can also be critical points.
- **Step 3:** **Verify Within Interval**: Ensure your critical points fall within the specified interval of the problem. For instance, if the interval is between 0 and \( 2\pi \), critical points like \( x = \frac{\pi}{2}, \pi, \text{and} \frac{3\pi}{2} \) are valid.

Identifying critical points paves the way for deeper analysis using subsequent tests, confirming their roles in establishing local maxima or minima.