Let `f (x) =ax ^(2) + bx+ c,a gt = and f (2-x) =f (2+x) AA x in R and f (x) =0` has 2 distinct real roots, then which of the following is true ?

Options

A. Atleast one roots must be positive
B. `f (2) lt f (0) gt f (1)`
C. Vertex of graph of `y =f (x)` is negative
D. Vertex of graph of `y =f (x)` lies in 3rd quadrat

Answer

To solve the problem, we start with the given function \( f(x) = ax^2 + bx + c \) where \( a > 0 \). We also know that \( f(2 - x) = f(2 + x) \) for all \( x \in \mathbb{R} \) and that \( f(x) = 0 \) has two distinct real roots.

### Step 1: Analyze the condition \( f(2 - x) = f(2 + x) \)

Substituting \( 2 - x \) and \( 2 + x \) into the function:

\[ f(2 - x) = a(2 - x)^2 + b(2 - x) + c \]
\[ = a(4 - 4x + x^2) + b(2 - x) + c \]
\[ = ax^2 - 4ax + 4a + 2b + c \]

Now, substituting \( 2 + x \):

\[ f(2 + x) = a(2 + x)^2 + b(2 + x) + c \]
\[ = a(4 + 4x + x^2) + b(2 + x) + c \]
\[ = ax^2 + 4ax + 4a + 2b + c \]

Setting these equal gives:

\[ ax^2 - 4ax + 4a + 2b + c = ax^2 + 4ax + 4a + 2b + c \]

### Step 2: Simplifying the equation

Subtracting \( ax^2 + 4a + 2b + c \) from both sides:

\[ -4ax = 4ax \]

This simplifies to:

\[ -8ax = 0 \]

Since \( a > 0 \), this implies \( x = 0 \). Thus, the function is symmetric about the line \( x = 2 \).

### Step 3: Roots of the function

Since \( f(x) \) has two distinct real roots, the discriminant must be positive:

\[ D = b^2 - 4ac > 0 \]

### Step 4: Analyze the vertex of the parabola

The vertex of the parabola given by \( f(x) \) is located at \( x = -\frac{b}{2a} \). Since the function is symmetric about \( x = 2 \), we have:

\[ -\frac{b}{2a} = 2 \implies b = -4a \]

### Step 5: Finding the value of \( f(0) \)

Now, we can find \( f(0) \):

\[ f(0) = a(0)^2 + b(0) + c = c \]

### Step 6: Determine the nature of the roots

Since the parabola opens upwards (as \( a > 0 \)) and has two distinct real roots, the vertex must be below the x-axis. Therefore, \( f(2) \) must also be negative.

### Conclusion: Evaluating the options

1. **At least one root must be positive**: True, since the roots are symmetric about \( x = 2 \).
2. **\( f(0) > f(2) \) and \( f(0) > f(1) \)**: True, because \( f(0) = c \) is above the x-axis while \( f(2) \) is below it.
3. **Vertex of the graph \( y = f(x) \) is negative**: True, since the vertex lies below the x-axis.
4. **Vertex lies in the third quadrant**: False, as the vertex lies in the fourth quadrant.

Thus, the correct options are 1, 2, and 3.