Let `f : [-3, 4] to R` such that `f ''(x) gt 0` for all `x in [-3,4],` then which of the following are always true ?

Options

A. f (x) has a relative minimum on `(-3, 4)`
B. f (x) has a minimum on `[-3,4]`
C. f (x) has a maximum on `[-3,4]`
D. if `f (3) =f (4),` then `f (x)` has a critical point on `[-3,4]`

Answer

To solve the problem, we need to analyze the implications of the second derivative \( f''(x) > 0 \) for all \( x \in [-3, 4] \). This condition indicates that the function \( f(x) \) is concave up on the entire interval, which affects the behavior of the first derivative \( f'(x) \).

### Step 1: Understanding the implications of \( f''(x) > 0 \)

Since \( f''(x) > 0 \), it means that the first derivative \( f'(x) \) is increasing on the interval \([-3, 4]\). Therefore, \( f(x) \) can either be strictly increasing or have a minimum point but cannot have a maximum point in the interval.

**Hint:** Remember that if the second derivative is positive, the first derivative is increasing, which indicates the function is either increasing or has a minimum.

### Step 2: Analyzing the options

1. **Option A:** \( f(x) \) has a relative minimum on \((-3, 4)\) (both excluded).
- This statement is not necessarily true. While the function is concave up, it could be strictly increasing without a relative minimum in the open interval. Thus, this option is **false**.

**Hint:** Consider the behavior of the function at the endpoints and whether it can have a minimum within the open interval.

2. **Option B:** \( f(x) \) has a minimum on \([-3, 4]\) (both included).
- This statement is true. Since the function is concave up, it must attain a minimum at one of the endpoints, either at \( f(-3) \) or \( f(4) \).

**Hint:** Think about the endpoints of the interval and where the minimum value can occur.

3. **Option C:** \( f(x) \) has a maximum on \([-3, 4]\) (both included).
- This statement is false. Since \( f''(x) > 0 \) indicates that the function is concave up, it cannot have a maximum in the interval.

**Hint:** Recall that a concave up function cannot have a maximum point; it can only have minimum points.

4. **Option D:** If \( f(3) = f(4) \), then \( f(x) \) has a critical point on \([-3, 4]\) (both included).
- This statement is true. If \( f(3) = f(4) \) and the function is concave up, then there must be a point where the function changes from decreasing to increasing, indicating a critical point exists.

**Hint:** Consider the implications of having equal function values at two points in a concave up function.

### Conclusion

From the analysis above, the true statements are:
- Option B: \( f(x) \) has a minimum on \([-3, 4]\) (both included).
- Option D: If \( f(3) = f(4) \), then \( f(x) \) has a critical point on \([-3, 4]\) (both included).

Thus, the correct options are **B and D**.