Question

Let \( a \in {R} \) and \( h \) be a positive real number. For any twice-differentiable function \( f : {R} \to {R} \), let \( P_f(x) \) be the interpolating polynomial of degree at most two that interpolates \( f \) at the points \( a - h, a, a + h \). Define \( d \) to be the largest integer such that any polynomial \( g \) of degree \( d \) satisfies \[ g''(a) = P_f''(a). \] The value of \( d \) is equal to (answer in integer):

Hint:

For interpolation problems, focus on the degree of the polynomial and the number of interpolation points.

Answer 1

Step 1: Degree of the interpolating polynomial. The polynomial \( P_f(x) \) is of degree at most two, interpolating \( f \) at three points. The second derivative \( P_f''(a) \) matches \( g''(a) \) if \( g \) is a polynomial of degree at most three. Step 2: Analyzing \( g(x) \). For \( g(x) \) of degree 3, the second derivative exists and matches \( P_f''(a) \). For \( g(x) \) of degree higher than 3, the condition does not hold. Step 3: Conclusion. The largest integer \( d \) is \( {3} \).