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A continuous and differentiable function...

A continuous and differentiable function `f(x)` is increasing in `(-oo,3/2)` and decreasing in `(3/2,oo)` then `x=3/2` is

A

point if local maxima

B

point of local minima

C

point of inflection

D

None of these

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the behavior of the function \( f(x) \) based on the information provided. ### Step-by-Step Solution: 1. **Understanding the behavior of the function**: We know that the function \( f(x) \) is continuous and differentiable. It is increasing in the interval \( (-\infty, \frac{3}{2}) \) and decreasing in the interval \( (\frac{3}{2}, \infty) \). 2. **Identifying critical points**: A point where the function changes from increasing to decreasing is a critical point. Here, \( x = \frac{3}{2} \) is such a point. 3. **Using the First Derivative Test**: Since \( f(x) \) is increasing in \( (-\infty, \frac{3}{2}) \), the derivative \( f'(x) > 0 \) in this interval. Conversely, since \( f(x) \) is decreasing in \( (\frac{3}{2}, \infty) \), the derivative \( f'(x) < 0 \) in this interval. 4. **Conclusion about the critical point**: At \( x = \frac{3}{2} \), the function transitions from increasing to decreasing. This indicates that \( x = \frac{3}{2} \) is a point of local maxima. 5. **Final answer**: Therefore, we conclude that \( x = \frac{3}{2} \) is a point of local maxima.
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