Consider a polynomial y = P(x) of the least degree passing through A(-1,1) and whose graph has two points of inflection B(1,2) and C with abscissa 0 at which the curve is inclined to the positive axis of abscissa at an angle of `sec^(-1)sqrt(2)`
The value of P(2) ius

Options

A. -1
B. `(-3)/(2)`
C. `(5)/(2)`
D. `(7)/(2)`

Explanation

To solve the problem, we need to find the polynomial \( P(x) \) of the least degree that passes through the point \( A(-1, 1) \) and has points of inflection at \( B(1, 2) \) and \( C(0, y_C) \) where the curve is inclined to the positive axis of abscissa at an angle of \( \sec^{-1} \sqrt{2} \).

### Step 1: Identify the degree of the polynomial
Since the polynomial has two points of inflection, it must be at least of degree 3. Thus, we can express \( P(x) \) in the form:
\[ P(x) = ax^3 + bx^2 + cx + d \]

### Step 2: Use the points of inflection
The points of inflection occur where the second derivative \( P''(x) = 0 \). We will find \( P''(x) \) and set it to zero at \( x = 0 \) and \( x = 1 \).

1. First, find the first derivative:
\[ P'(x) = 3ax^2 + 2bx + c \]
2. Then, find the second derivative:
\[ P''(x) = 6ax + 2b \]
3. Set \( P''(0) = 0 \):
\[ 2b = 0 \implies b = 0 \]
4. Set \( P''(1) = 0 \):
\[ 6a(1) + 2b = 0 \implies 6a = 0 \implies a = 0 \]

### Step 3: Update the polynomial
Since \( a = 0 \) and \( b = 0 \), we can simplify \( P(x) \):
\[ P(x) = cx + d \]

### Step 4: Use the point A(-1, 1)
We know that \( P(-1) = 1 \):
\[ c(-1) + d = 1 \implies -c + d = 1 \quad \text{(Equation 1)} \]

### Step 5: Use the point B(1, 2)
We also know that \( P(1) = 2 \):
\[ c(1) + d = 2 \implies c + d = 2 \quad \text{(Equation 2)} \]

### Step 6: Solve the equations
Now we have two equations:
1. \( -c + d = 1 \)
2. \( c + d = 2 \)

Subtract Equation 1 from Equation 2:
\[ (c + d) - (-c + d) = 2 - 1 \implies 2c = 1 \implies c = \frac{1}{2} \]

Substituting \( c \) back into Equation 2:
\[ \frac{1}{2} + d = 2 \implies d = 2 - \frac{1}{2} = \frac{3}{2} \]

### Step 7: Write the polynomial
Now we have:
\[ P(x) = \frac{1}{2}x + \frac{3}{2} \]

### Step 8: Find \( P(2) \)
Now we can find \( P(2) \):
\[ P(2) = \frac{1}{2}(2) + \frac{3}{2} = 1 + \frac{3}{2} = \frac{5}{2} \]

Thus, the value of \( P(2) \) is \( \frac{5}{2} \).

### Final Answer
\[ \boxed{\frac{5}{2}} \]