I tried this question 5 times already and still wrong, will appreciate any help. **Understanding Derivatives and Function Behavior****Question:**Suppose the derivative of \( f \) is \( f'(x) = (x + 1)^4 (x - 2)^3 (x - 3)^6 \). On what interval is \( f \) increasing? (Enter your answer in interval notation.)**Student's Response:**\((2, 3) \cup (3, \infty)\)**Feedback:**IncorrectTo determine where the function \( f \) is increasing, we need to analyze the sign of its derivative, \( f'(x) \), given by:\[ f'(x) = (x + 1)^4 (x - 2)^3 (x - 3)^6 \]Here is a step-by-step analysis:1. **Identify Critical Points:** - The derivative \( f'(x) \) is zero whenever \( x = -1 \), \( x = 2 \), or \( x = 3 \).2. **Sign Analysis:** - For \( x < -1 \): All factors are negative or positive \( \implies f'(x) < 0 \) (since \( (x + 1)^4 \) is always positive). - For \( -1 < x < 2 \): \( (x + 1)^4 > 0 \), \( (x - 2)^3 < 0 \), \( (x - 3)^6 > 0 \) \( \implies f'(x) < 0 \). - For \( 2 < x < 3 \): \( (x + 1)^4 > 0 \), \( (x - 2)^3 > 0 \), \( (x - 3)^6 > 0 \) \( \implies f'(x) > 0 \). - For \( x > 3 \): \( (x + 1)^4 > 0 \), \( (x - 2)^3 > 0 \), \( (x - 3)^6 > 0 \) \( \implies f'(x) > 0 \).3. **Conclusion:** - \( f \) is increasing where \( f'(x) > 0 \). From our analysis, this occurs on the intervals \( (2, 3) \) and \(

Name: I tried this question 5 times already and still wrong, will appreciate
Uploaded: 2024-03-20T07:06:16.000Z
Channel: Bartleby
Description: I tried this question 5 times already and still wrong, will appreciate any help. **Understanding Derivatives and Function Behavior****Question:**Suppose the derivative of \( f \) is \( f'(x) = (x + 1)^4 (x - 2)^3 (x - 3)^6 \). On what interval is \(