Let `f (x) =x ^(2) -4x +c AA x in R,` where c is a real constant, then which of the following is/are true ?

Options

A. `f (0) gt f (1) gt f (2)`
B. `f (2) gt f (3) gt f(4)`
C. `f (1) lt f (4) lt f (-1)`
D. `f (0) =f (4) gt f(3)`

Answer

To solve the problem, we need to analyze the function \( f(x) = x^2 - 4x + c \) and determine the truth of the given options based on its properties.

### Step-by-Step Solution:

1. **Identify the function and its derivative**:
\[ f(x) = x^2 - 4x + c \]
The derivative of the function is:
\[ f'(x) = 2x - 4 \]

**Hint**: The derivative helps us find critical points where the function may have maxima or minima.

2. **Find critical points by setting the derivative to zero**:
\[ 2x - 4 = 0 \implies x = 2 \]

**Hint**: Critical points are found where the derivative equals zero.

3. **Determine the nature of the critical point**:
To determine whether \( x = 2 \) is a maximum or minimum, we can use the second derivative test or analyze the sign of the first derivative around \( x = 2 \).

- If we choose a value less than 2 (e.g., \( x = 1 \)):
\[ f'(1) = 2(1) - 4 = -2 \quad (\text{negative, function is decreasing}) \]
- If we choose a value greater than 2 (e.g., \( x = 3 \)):
\[ f'(3) = 2(3) - 4 = 2 \quad (\text{positive, function is increasing}) \]

Since \( f'(x) \) changes from negative to positive at \( x = 2 \), this indicates that \( x = 2 \) is a point of minimum.

**Hint**: The sign of the derivative tells us whether the function is increasing or decreasing.

4. **Evaluate the function at specific points to analyze options**:
- For \( x = 0 \):
\[ f(0) = 0^2 - 4(0) + c = c \]
- For \( x = 1 \):
\[ f(1) = 1^2 - 4(1) + c = c - 3 \]
- For \( x = 3 \):
\[ f(3) = 3^2 - 4(3) + c = 9 - 12 + c = c - 3 \]
- For \( x = 4 \):
\[ f(4) = 4^2 - 4(4) + c = 16 - 16 + c = c \]
- For \( x = -1 \):
\[ f(-1) = (-1)^2 - 4(-1) + c = 1 + 4 + c = 5 + c \]

**Hint**: Evaluating the function at specific points helps us understand its behavior.

5. **Analyze the results**:
- From the evaluations:
- \( f(0) = c \)
- \( f(1) = c - 3 \)
- \( f(3) = c - 3 \)
- \( f(4) = c \)
- \( f(-1) = 5 + c \)

We can see that:
- \( f(0) = f(4) \)
- \( f(1) = f(3) < f(0) = f(4) \)
- \( f(-1) > f(0) = f(4) \)

This indicates that the function is decreasing from \( x = 0 \) to \( x = 1 \), reaching a minimum at \( x = 2 \), and then increasing thereafter.

6. **Conclusion**:
Based on the analysis, we can conclude:
- Option A is true (function decreases until \( x = 2 \)).
- Option B is false (function does not decrease for \( x \geq 2 \)).
- Option C is true (function increases and then decreases).
- Option D is true (function has the same value at \( x = 0 \) and \( x = 4 \), with a minimum at \( x = 2 \)).

Therefore, the final answer is:
- **Correct options**: A, C, D.