let f(x) be a function such that its derivative is given by: f'(x)=5(x-1)^2(x-2)^3(x-3)^4. which statement is true about the function f(x)? ### Analyzing the Derivative of a Function#### Given ProblemLet \( f(x) \) be a function such that its derivative is given by:\[ f'(x) = 5(x-1)^2 (x-2)^3 (x-3)^4. \]Which statement is true about the function \( f(x) \)?- \(\bigcirc \) It has three local max or local min at \( x = 1, x = 2, \) and \( x = 3 \)- \(\bigcirc \) It has a local min at \( x = 2 \)- \(\bigcirc \) It has a local min at \( x = 3 \)- \(\bigcirc \) It has no local max nor local min- \(\bigcirc \) None of the above#### Detailed AnalysisTo determine the local extrema of \( f(x) \), we need to analyze its derivative \( f'(x) \).1. **Critical Points:** - The derivative \( f'(x) = 0 \) at \( x = 1, 2, 3 \).2. **Behavior of \( f'(x) \) Based on its Factors:** - **At \( x = 1 \):** The factor \((x-1)^2\) makes \( f'(x) \) equal zero. Since the power is even (2), the sign of \( f'(x) \) does not change around \( x = 1 \). This suggests that \( x = 1 \) is more likely a point of inflection rather than a local maximum or minimum. - **At \( x = 2 \):** The factor \((x-2)^3\) makes \( f'(x) \) equal zero. Since the power is odd (3), the sign of \( f'(x) \) changes around \( x = 2 \). This indicates that \( x = 2 \) is a local extremum. Given that the factor is raised to an odd power greater than 1, which affects the steepness at that point, it suggests a local minimum. - **At \( x = 3 \):** The factor \((x-3)^4\) makes \( f'(x) \) equal zero. Since the power is even (4),

Name: let f(x) be a function such that its derivative is given by: f'(x)=5(x
Uploaded: 2024-03-20T06:47:22.000Z
Channel: Bartleby
Description: let f(x) be a function such that its derivative is given by: f'(x)=5(x-1)^2(x-2)^3(x-3)^4. which statement is true about the function f(x)? ### Analyzing the Derivative of a Function#### Given ProblemLet \( f(x) \) be a function such that its derivat