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If f'' (x) lt 0,x in (a,b), f(c ) attain...

If `f'' (x) lt 0,x in (a,b), f(c )` attains maximum value at `(c,f(c ))`, where a < c < b, then f'(c ) is equal to :

A

`(f(b) - f(a))/(2)`

B

`f'(c ) = 0`

C

`f'( c) = (f(a) -f(b))/(2)`

D

`f'(c ) = 2(f(b) - f(a))`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the conditions given in the question regarding the function \( f \) and its derivatives. ### Step-by-Step Solution: 1. **Understanding the Condition**: We are given that \( f''(x) < 0 \) for \( x \in (a, b) \). This indicates that the function \( f \) is concave down in the interval \( (a, b) \). **Hint**: Remember that if the second derivative is negative, the function is concave down, which suggests that any critical point in this interval is a maximum. 2. **Identifying the Maximum**: The function \( f \) attains its maximum value at the point \( (c, f(c)) \), where \( a < c < b \). At this point \( c \), the function reaches its highest value in the interval. **Hint**: A maximum point is where the function changes from increasing to decreasing. 3. **Applying the First Derivative Test**: At a maximum point, the first derivative \( f'(c) \) must be equal to zero. This is because the slope of the tangent line to the curve at the maximum point is horizontal. **Mathematical Expression**: \[ f'(c) = 0 \] **Hint**: The first derivative gives the slope of the tangent line. At a maximum, this slope is zero. 4. **Conclusion**: Since we have established that \( f'(c) = 0 \) at the point where \( f(c) \) is maximum, we conclude that: \[ f'(c) = 0 \] **Final Answer**: Therefore, \( f'(c) \) is equal to \( 0 \).
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