Consider that ` f(x) =ax^(2) + bx +c, D = b^(2)-4ac` , then which of the following is not true ?

Options

A. If `a gt 0` then minimum value of f(x) is `(-D)/(4a)`
B. If `a lt 0`, then maximum value of f(x) is `(-D)/(4a)`
C. if `a gt 0,D lt 0`, then `f(x) gt 0` for all x ` in ` R
D. If `a gt 0, D gt 0` , then `f(x) gt 0` for alll ` x in R`

Answer

To solve the problem, we need to analyze the properties of the quadratic function \( f(x) = ax^2 + bx + c \) and its discriminant \( D = b^2 - 4ac \). We will evaluate the statements regarding the function and identify which one is not true.

### Step-by-Step Solution:

1. **Understanding the Quadratic Function**:
The function \( f(x) = ax^2 + bx + c \) is a quadratic function. The shape of the graph (parabola) depends on the coefficient \( a \).

2. **Discriminant Analysis**:
The discriminant \( D = b^2 - 4ac \) helps us determine the nature of the roots of the quadratic equation:
- If \( D > 0 \): The quadratic has two distinct real roots.
- If \( D = 0 \): The quadratic has exactly one real root (a repeated root).
- If \( D < 0 \): The quadratic has no real roots (the parabola does not intersect the x-axis).

3. **Case 1: \( a > 0 \)**:
- The parabola opens upwards.
- If \( D < 0 \), the vertex of the parabola is the minimum point, and the function value \( f(x) \) is always greater than 0 for all real \( x \).

4. **Case 2: \( a < 0 \)**:
- The parabola opens downwards.
- If \( D < 0 \), the vertex is the maximum point, and the function value \( f(x) \) is always less than 0 for all real \( x \).

5. **Evaluating the Statements**:
- **Statement 1**: When \( a > 0 \) and \( D < 0 \), \( f(x) \) is always positive. (True)
- **Statement 2**: When \( a < 0 \) and \( D < 0 \), \( f(x) \) is always negative. (True)
- **Statement 3**: When \( a > 0 \) and \( D > 0 \), the function has two real roots and will cross the x-axis. (True)
- **Statement 4**: When \( a > 0 \) and \( D > 0 \), the function value \( f(x) \) can be less than 0 between the roots. (False)

6. **Conclusion**:
The statement that is not true is **Statement 4**. When \( a > 0 \) and \( D > 0 \), the function will cross the x-axis, and there will be intervals where \( f(x) \) is indeed less than 0, specifically between the two roots.

### Final Answer:
The statement that is not true is **Statement 4**.