Lets have first derivative at c such that f'(c) = 0 and f'(c) `gt 0, AA x in (c- delta c)`, also f'(c)`gt 0, AA x in (c,c+delta)`, then c is a point of

Options

A. Local maximum
B. Local minimum
C. Point of inflection
D. Absolute maximum

Answer

To solve the given problem step by step, we need to analyze the conditions provided about the function \( f \) and its first derivative \( f' \).

### Step 1: Understand the Given Conditions
We are given that:
1. \( f'(c) = 0 \)
2. \( f'(x) > 0 \) for \( x \) in the interval \( (c - \delta, c) \)
3. \( f'(x) > 0 \) for \( x \) in the interval \( (c, c + \delta) \)

### Step 2: Analyze the First Derivative
From the first condition, \( f'(c) = 0 \), we know that at the point \( c \), the slope of the tangent to the curve \( f(x) \) is horizontal. This implies that \( c \) could be a local maximum, local minimum, or a point of inflection.

### Step 3: Determine Behavior on the Left Side of \( c \)
Since \( f'(x) > 0 \) for \( x \) in \( (c - \delta, c) \), it indicates that the function \( f(x) \) is increasing as we approach \( c \) from the left. Therefore, as \( x \) gets closer to \( c \) from the left, the value of \( f(x) \) is increasing.

### Step 4: Determine Behavior on the Right Side of \( c \)
Similarly, since \( f'(x) > 0 \) for \( x \) in \( (c, c + \delta) \), it indicates that the function \( f(x) \) is also increasing as we move away from \( c \) to the right. Thus, as \( x \) gets closer to \( c \) from the right, the value of \( f(x) \) is also increasing.

### Step 5: Conclusion about the Point \( c \)
Since \( f'(c) = 0 \) and the function is increasing on both sides of \( c \), we can conclude that \( c \) is neither a local maximum nor a local minimum. Instead, it is a point where the function changes from increasing to increasing, which characterizes a point of inflection.

### Final Answer
Thus, \( c \) is a point of inflection.

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