If f(c) `lt` 0 and f'(c) `gt` 0 , then at x = c , f (x) is -

Options

A. maximum
B. minimum
C. neither maximum nor minimum
D. either maximum or minimum

Answer

To solve the problem, we need to analyze the given conditions regarding the function \( f(x) \) at the point \( x = c \).

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- We have \( f(c) < 0 \): This means that the value of the function at \( x = c \) is negative.
- We also have \( f'(c) > 0 \): This indicates that the derivative of the function at \( x = c \) is positive, which implies that the function is increasing at that point.

2. **Analyzing the Conditions**:
- The condition \( f(c) < 0 \) tells us that the function is below the x-axis at \( x = c \).
- The condition \( f'(c) > 0 \) means that as we move to the right of \( c \), the function values are increasing.

3. **Determining Maxima or Minima**:
- For a function to have a local maximum at a point \( c \), we typically expect \( f'(c) = 0 \) and \( f'(x) < 0 \) for \( x < c \) and \( f'(x) > 0 \) for \( x > c \). However, since \( f'(c) > 0 \), this indicates that the function is not at a maximum.
- For a function to have a local minimum at a point \( c \), we usually expect \( f'(c) = 0 \) and \( f'(x) > 0 \) for \( x < c \) and \( f'(x) < 0 \) for \( x > c \). Again, since \( f'(c) > 0 \), this indicates that the function is not at a minimum.

4. **Conclusion**:
- Since neither condition for a local maximum nor a local minimum is satisfied at \( x = c \), we conclude that at \( x = c \), the function \( f(x) \) is neither a maximum nor a minimum.
- Therefore, the answer is: **neither maximum nor minimum**.

### Final Answer:
At \( x = c \), \( f(x) \) is **neither maximum nor minimum**.

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