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f(x) is increasing on interval and f(x) is decreasing on interval

The graph provided is of the derivative \( f'(x) = (x + 3)(x - 1) \). To determine where the original function \( f(x) \) is increasing or decreasing, we analyze the behavior of its derivative.

- **\( f(x) \) is increasing** when \( f'(x) > 0 \) (i.e., where the derivative is positive).
- **\( f(x) \) is decreasing** when \( f'(x) < 0 \) (i.e., where the derivative is negative).

### Step-by-Step Analysis:
1. The derivative \( f'(x) = (x + 3)(x - 1) \) is a quadratic function. Its roots occur when \( f'(x) = 0 \), which gives:
   - \( x + 3 = 0 \) → \( x = -3 \)
   - \( x - 1 = 0 \) → \( x = 1 \)

2. This quadratic opens upwards (since the coefficient of \( x^2 \) is positive), so the graph is negative between the roots and positive outside the roots.

3. Based on the graph and the nature of the quadratic:
   - \( f(x) \) is **decreasing** on the interval \( (-\infty, -3) \) because \( f'(x) < 0 \).
   - \( f(x) \) is **increasing** on the intervals \( (-3, 1) \) and \( (1, \infty) \) because \( f'(x) > 0 \).

### Final Answer:
- \( f(x) \) is increasing on \( (-3, 1) \cup (1, \infty) \).
- \( f(x) \) is decreasing on \( (-\infty, -3) \).

Would you like further details or explanations?

### Related Questions:
1. How do you determine concavity using the second derivative?
2. What is the significance of critical points for \( f(x) \)?
3. How does the graph of \( f'(x) \) relate to the turning points of \( f(x) \)?
4. Can you explain how to find inflection points from the second derivative?
5. What does it mean when the derivative of a function is zero?

### Tip:
When interpreting the graph of a derivative, remember that positive values indicate an increasing original function and negative values indicate a decreasing original function.

The graph of the derivative f'(x) = (x + 3)(x - 1) appears below. Find where f(x) is increasing and decreasing.

Analyze the graph of the derivative f'(x) = (x + 3)(x - 1) to determine where the original function f(x) is increasing and decreasing. This problem covers key calculus concepts such as identifying intervals of monotonicity based on the derivative.

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