https://file.aiquickdraw.com/mathbot/file/1727796021554sgqgnzgk.jpg

To answer the questions based on the graph:

### Part (a) Find the values of \( x \) where \( F'(x) = 0 \):
The derivative \( F'(x) = 0 \) at points where the function \( F(x) \) has horizontal tangents, meaning the slope of the tangent line is zero. These points correspond to the peaks and troughs on the graph. From the graph, this occurs at points \( x = n, r, t \), where the curve has local maxima or minima.

So, the answer is:
**n, r, t**

### Part (b) Where is \( F'(x) > 0 \) (increasing)?:
The function \( F'(x) > 0 \) when \( F(x) \) is increasing, which happens between points where the graph goes upwards (positive slope). Based on the graph:
- \( F'(x) > 0 \) from \( m \) to \( n \) (as the curve rises) and from \( s \) to \( t \) (where it rises again).

Thus, the answer in interval notation is:
**(m, n), (s, t)**

Do you want further details or have any questions?

Here are 5 related questions:
1. How can we determine if a function has a local maximum or minimum using the first derivative?
2. What does \( F'(x) < 0 \) indicate about the function?
3. How do you find points of inflection using a graph?
4. What is the relationship between the second derivative and concavity?
5. How does the sign of the first derivative relate to the behavior of the original function?

**Tip**: Remember that the first derivative gives the slope of the tangent to the curve at each point and can help identify intervals of increasing or decreasing behavior!

The graph of the function F(x) is given. The problem asks to find where F'(x) = 0 and where F'(x) > 0.

Analyze the graph of a function to find where F'(x) = 0, indicating critical points, and where F'(x) > 0, indicating increasing behavior. This problem is ideal for students in Grades 11-12 studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation