Number 9 **Educational Website Content**---### Understanding Derivatives and Critical Points**Exercise 5-7**- **5.** \( f(x) = \frac{2}{3}x^3 - x^2 \)- **7a.** On the basis of your answers to Exercises 1–6, describe how the first derivative of a function could be used to determine whether a local maximum or a local minimum occurs at a critical number.- **7b.** If \( c \) is a critical number, is it true that \( (c, f(c)) \) is either a local maximum or minimum point?---### Graph Analysis**8.** The graph of a function \( f \) is shown below with several points labeled.#### Diagram Explained:- **Graph Description:** A curve with labeled points A, B, C, and D.**Questions:**- **8a.** At which labeled points is \( f' \) positive? Explain.- **8b.** Between which pairs of labeled points does \( f \) have a critical point?- **8c.** Between which pairs of labeled points is \( f' \) increasing?- **8d.** Between which pairs of labeled points does \( f' \) achieve its minimum value?---### Graph Sketching**9.** - **9a.** Sketch the graph of a function \( f \) for which \( f(x) > 0 \) and \( f'(x) > 0 \) for all \( x \).- **9b.** Sketch the graph of a function \( g \) for which \( g(x) > 0 \) and \( g'(x) < 0 \) for all \( x \).- **9c.** Give formulas for functions with these properties.---### Numerical Derivatives**10.** The numerical derivative \( \text{nDeriv} \) frequently calculates misleading values at critical numbers where the derivative is undefined. For example, let \( f(x) = \sqrt[3]{x - 2} \) and \( g(x) = \sqrt[3]{(x - 2)^2} \). - **Example Explained:** - At \( x = 2 \), both \( f'(2) \) and \( g'(2) \) do not exist. -

Name: Number 9 **Educational Website Content**---### Understanding Derivativ
Uploaded: 2024-03-05T11:52:21.000Z
Channel: Bartleby
Description: Number 9 **Educational Website Content**---### Understanding Derivatives and Critical Points**Exercise 5-7**- **5.** \( f(x) = \frac{2}{3}x^3 - x^2 \)- **7a.** On the basis of your answers to Exercises 1–6, describe how the first derivative of a func