Let `f(x)=ax^(2)+bx+c` such that `f(1)=f(-1)` and a,b,c are in Arithmetic Progression (AP).
Q. What is the value of b?

Options

A. `-1`
B. `0`
C. 1
D. Cannot be determined

Answer

To solve the problem, we need to analyze the function \( f(x) = ax^2 + bx + c \) under the conditions given.

### Step 1: Set up the equations based on the conditions
We know that \( f(1) = f(-1) \). Let's calculate both:

\[ f(1) = a(1)^2 + b(1) + c = a + b + c \]

\[ f(-1) = a(-1)^2 + b(-1) + c = a - b + c \]

### Step 2: Set the equations equal to each other
Since \( f(1) = f(-1) \), we can set the two expressions equal to each other:

\[ a + b + c = a - b + c \]

### Step 3: Simplify the equation
We can simplify this equation by subtracting \( a \) and \( c \) from both sides:

\[ b = -b \]

### Step 4: Solve for \( b \)
Now, we can add \( b \) to both sides:

\[ b + b = 0 \]

This simplifies to:

\[ 2b = 0 \]

Dividing both sides by 2 gives:

\[ b = 0 \]

### Step 5: Verify the condition of Arithmetic Progression
We also know that \( a, b, c \) are in Arithmetic Progression (AP). For \( a, b, c \) to be in AP, the condition is:

\[ 2b = a + c \]

Since we found \( b = 0 \), substituting gives:

\[ 2(0) = a + c \implies 0 = a + c \]

This means \( a = -c \), which is consistent with the AP condition.

### Conclusion
Thus, the value of \( b \) is:

\[ \boxed{0} \]

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