f(c) is a minimum value of f(x) if -

Options

A. `f'(c) = 0 , f'' (c) gt 0`
B. `f'(c) = 0 , f'' (c) lt 0`
C. `f ' (c) ne 0 , f''(c) = 0`
D. `f '(c) lt 0 , f ''(c) gt 0`

Answer

To determine the conditions under which \( f(c) \) is a minimum value of \( f(x) \), we can follow these steps:

### Step 1: Find the First Derivative
To find the critical points of the function \( f(x) \), we need to compute the first derivative \( f'(x) \).

### Step 2: Set the First Derivative to Zero
We set \( f'(c) = 0 \) to find the critical points. This indicates that the slope of the tangent to the curve at point \( c \) is zero, suggesting a potential minimum or maximum.

### Step 3: Determine the Nature of the Critical Point
To determine whether the critical point \( c \) is a minimum, we need to examine the second derivative \( f''(x) \).

### Step 4: Compute the Second Derivative
Calculate \( f''(c) \).

### Step 5: Apply the Second Derivative Test
- If \( f''(c) > 0 \), then \( f(c) \) is a local minimum.
- If \( f''(c) < 0 \), then \( f(c) \) is a local maximum.
- If \( f''(c) = 0 \), the test is inconclusive.

### Conclusion
Thus, \( f(c) \) is a minimum value of \( f(x) \) if:
1. \( f'(c) = 0 \)
2. \( f''(c) > 0 \)

### Summary of Conditions
The conditions for \( f(c) \) to be a minimum are:
- \( f'(c) = 0 \)
- \( f''(c) > 0 \)