# **Exercise Questions** All questions in this exercise are listed below. Click on a question to view its solution. --- # **Key Concepts** This exercise focuses on the following concepts: * **Critical Values:** Identifying points where the first derivative is zero or undefined. * **Absolute Extrema:** Finding the highest and lowest values of a function on a closed interval. * **Concavity:** Using the second derivative to determine if a graph is concave upward or downward. * **Points of Inflection:** Identifying points where the concavity of a function changes. * **Second Derivative Test:** A method for classifying relative extrema using the sign of the second derivative at critical points. --- # **Important Formulas** Below are the key formulas used in this exercise: | Concept | Condition | | :--- | :--- | | **Critical Value** | $f'(c) = 0$ or $f'(c)$ is undefined | | **Concave Upward** | $f''(x) > 0$ | | **Concave Downward** | $f''(x) < 0$ | | **Inflection Point** | $f''(x) = 0$ or undefined (and concavity changes) | | **Relative Minimum** | $f'(c) = 0$ and $f''(c) > 0$ | | **Relative Maximum** | $f'(c) = 0$ and $f''(c) < 0$ | --- # **Summary** This exercise covers the fundamental applications of the first and second derivatives in curve sketching and optimization. Key learnings include the systematic identification of critical points and the use of the second derivative to determine the shape of a graph (concavity) and the nature of its extrema. Strategies involve evaluating functions at endpoints for absolute extrema and applying the Second Derivative Test as an efficient alternative to the First Derivative Test for classifying relative maxima and minima.