GATE 2024 MA – Question 32
P_f is the degree-at-most-two interpolant to f at a−h,a,a+h, h>0. Find the largest degree d for which every polynomial g of degree d satisfies g″(a)=P_g″(a).
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Correct answer: 3 to 3
Explanation
The interpolant’s second derivative is $[g(a-h)-2g(a)+g(a+h)]/h^2$. This central formula is exact through cubic degree because odd powers cancel and the quadratic term yields g″(a). For g=(x−a)⁴ it gives 2h² while g″(a)=0. Hence **d=3**.