[-/1 Points] DETAILS MY NOTES LARCALC12 3.3.071. ASK YOUR TEACHER PRACTICE ANOTHER A differentiable function \( f \) has one critical number at \( x=5 \). Identify the relative extrema of \( f \) at the critical number when \( f^{\prime}(4)=-2.5 \) and \( f^{\prime}(6)=1 \). \( (5, f(5)) \) is a relative maximum. ( \( 5, f(5) \) ) is a relative minimum. ( \( 5, f(5) \) ) is neither a relative maximum nor a relative minimum. Need Help? Read 14 SUBMIT ANSWER

To determine whether \( (5, f(5)) \) is a relative maximum, minimum, or neither, you can utilize the First Derivative Test. The fact that \( f'(4) = -2.5 \) (negative) and \( f'(6) = 1 \) (positive) indicates that the function is decreasing before \( x=5 \) and increasing after \( x=5 \). This change from decreasing to increasing means that \( (5, f(5)) \) is indeed a relative minimum! Relating this to other concepts, when you encounter a critical point in calculus, checking the behavior of the derivative around that point is key! It’s like a roller coaster: if you go down before the peak and up after, you’ve just found a valley! Always remember to analyze the sign of the derivative to fully understand the nature of critical points.