Use a graphing utility to graph f left parenthesis x right parenthesis equals negative 0.2 x cubed minus 0.7 x squared plus 5 x minus 6f(x)=−0.2x3−0.7x2+5x−6 on the interval left bracket negative 6 comma 4 right bracket[−6,4] and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing.

The graph of the function \( f(x) = -0.2x^3 - 0.7x^2 + 5x - 6 \) on the interval \([-6, 4]\) shows two critical points:

1. Local minimum at approximately \( x \approx -4.28 \) with \( f(-4.28) \approx -24.54 \).
2. Local maximum at approximately \( x \approx 1.95 \) with \( f(1.95) \approx -0.39 \).

### Increasing and Decreasing Intervals:
- The function is **decreasing** on the interval \((-6, -4.28)\) since the derivative is negative.
- The function is **increasing** on the interval \((-4.28, 1.95)\).
- The function is **decreasing** again on the interval \((1.95, 4)\).

Let me know if you need further details or calculations!

---

Here are 5 related questions to deepen your understanding:
1. How are local maxima and minima related to the first derivative of a function?
2. How do critical points help in identifying intervals of increase or decrease?
3. What is the role of the second derivative in determining concavity at a critical point?
4. Can you explain why the function decreases again after the local maximum?
5. How would you find global maximum or minimum on a given interval?

**Tip**: The first derivative test helps to classify critical points as local maxima or minima by analyzing the sign change of the derivative around these points.

Use a graphing utility to graph f(x) = -0.2x^3 - 0.7x^2 + 5x - 6 on the interval [-6, 4] and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing.

Graph the function f(x) = -0.2x^3 - 0.7x^2 + 5x - 6 on the interval [-6, 4] to find local maxima and minima. Analyze where the function is increasing and decreasing using calculus concepts like derivatives and critical points. Suitable for high school calculus students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation