If `f''(x) lt 0 AA x in (a,b)` and `(c, f(c))` is a point lying on the curve `y =f(x)`, where `a < c < b` and for that value of `c`, `f(c)` has a maximum then `f'(c)` equals

Options

A. `(f(b) - f(a))/b-a)`
B. `2/(b-a) {f(b) - f(a)}`
C. `2f(b) -f(a)/2b-a`
D. 0

Answer

To solve the problem, we need to analyze the given conditions step by step.

### Step-by-step Solution:

1. **Understanding the Given Information**:
We are given that \( f''(x) < 0 \) for all \( x \) in the interval \( (a, b) \). This indicates that the function \( f(x) \) is concave down on this interval.

**Hint**: Remember that if the second derivative is negative, the graph of the function is bending downwards.

2. **Identifying the Point \( c \)**:
We know that \( (c, f(c)) \) is a point on the curve where \( c \) lies in the interval \( (a, b) \) and \( f(c) \) is a maximum.

**Hint**: A maximum point on a curve implies that the function is not increasing at that point.

3. **Applying the First Derivative Test**:
Since \( f(c) \) is a maximum, we can apply the first derivative test. At a maximum point, the first derivative \( f'(c) \) must be equal to 0. This is because the slope of the tangent line at a maximum point is horizontal.

**Hint**: Recall that at local maxima or minima, the first derivative equals zero.

4. **Conclusion**:
Therefore, we conclude that:
\[ f'(c) = 0 \]

### Final Answer:
The value of \( f'(c) \) is \( 0 \).

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