Let `f(x) = ax^2 + bx +C,a,b,c in R`.It is given `|f(x)|<=1,|x|<=1` The possible value of `|a + c|` ,if `8/3a^2+2b^2` is maximum, is given by

Options

A. 1
B. 0
C. 2
D. 3

Answer

To solve the problem, we need to analyze the quadratic function \( f(x) = ax^2 + bx + c \) under the constraints given. The goal is to find the possible value of \( |a + c| \) when \( \frac{8}{3}a^2 + 2b^2 \) is maximized, given that \( |f(x)| \leq 1 \) for \( |x| \leq 1 \).

### Step 1: Understand the constraints
We know that:
1. \( |f(x)| \leq 1 \) for \( |x| \leq 1 \)
2. This means \( f(1) \) and \( f(-1) \) must also satisfy this condition.

Calculating \( f(1) \) and \( f(-1) \):
- \( f(1) = a + b + c \)
- \( f(-1) = a - b + c \)

From the constraints, we have:
\[ |a + b + c| \leq 1 \quad \text{(1)} \]
\[ |a - b + c| \leq 1 \quad \text{(2)} \]

### Step 2: Set up the equations
From equations (1) and (2), we can derive:
1. \( -1 \leq a + b + c \leq 1 \)
2. \( -1 \leq a - b + c \leq 1 \)

### Step 3: Analyze the expressions
We can express these inequalities in terms of \( c \):
- From (1): \( -1 - a - b \leq c \leq 1 - a - b \)
- From (2): \( -1 - a + b \leq c \leq 1 - a + b \)

### Step 4: Find the maximum of \( \frac{8}{3}a^2 + 2b^2 \)
To maximize \( \frac{8}{3}a^2 + 2b^2 \), we can use the method of Lagrange multipliers or analyze the boundary conditions given by the inequalities.

### Step 5: Substitute values
Assuming \( b = 0 \) for simplicity, we can analyze:
\[ \frac{8}{3}a^2 \quad \text{with} \quad |a + c| \leq 1 \]
If \( b = 0 \), then:
- \( |a + c| \leq 1 \) implies \( c = -a \) when maximizing \( |a + c| \).

Substituting \( c = -a \):
\[ |a - a| = |0| \quad \text{(not maximizing)} \]
Instead, we need to find \( c \) such that \( a + c \) is maximized.

### Step 6: Solve for \( c \)
From the previous analysis, we can set \( c = 1 - a \) or \( c = -1 - a \) to find the maximum value of \( |a + c| \):
- If \( c = 1 - a \), then \( |a + c| = |1| = 1 \)
- If \( c = -1 - a \), then \( |a + c| = |-1| = 1 \)

### Conclusion
Thus, the maximum value of \( |a + c| \) when \( \frac{8}{3}a^2 + 2b^2 \) is maximized is:
\[ \boxed{1} \]