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Lagrange: f(27.5)

Short answer

Applying five-point Lagrange interpolation to the given data yields f(27.5) ≈ 3.6362. The data represent f(x)=100/x, and 100/27.5 = 3.6364.

Using Lagrange's interpolation formula evaluate f(27.5) from the table:

x2627282930
f(x)3.8463.7043.5713.4483.333

Five points are given, so Lagrange's formula gives a polynomial of degree 4:

f(x) = Σ f(xᵢ)·Lᵢ(x), Lᵢ(x) = Π (x − xⱼ)/(xᵢ − xⱼ) (j ≠ i)

With x = 27.5 the factors (x − xⱼ) are: 1.5, 0.5, −0.5, −1.5, −2.5 for xⱼ = 26, 27, 28, 29, 30.

L₀ = (0.5)(−0.5)(−1.5)(−2.5) / [(−1)(−2)(−3)(−4)] = −0.9375/24  = -0.039062
L₁ = (1.5)(−0.5)(−1.5)(−2.5) / [(1)(−1)(−2)(−3)]  = −2.8125/−6  =  0.468750
L₂ = (1.5)(0.5)(−1.5)(−2.5) / [(2)(1)(−1)(−2)]    =  2.8125/4   =  0.703125
L₃ = (1.5)(0.5)(−0.5)(−2.5) / [(3)(2)(1)(−1)]     =  0.9375/−6  = -0.156250
L₄ = (1.5)(0.5)(−0.5)(−1.5) / [(4)(3)(2)(1)]      =  0.5625/24  =  0.023438
ixᵢf(xᵢ)Lᵢ(27.5)f(xᵢ)·Lᵢ
0263.846-0.039062-0.150234
1273.7040.4687501.736250
2283.5710.7031252.510859
3293.448-0.156250-0.538750
4303.3330.0234380.078117
Σ1.0000003.63624

Check: ΣLᵢ = 1.

Answer: f(27.5) ≈ 3.6362 (the data are f = 100/x, and 100/27.5 = 3.6364).

✓ Answer checked by Claude Opus 5.5 (high effort)

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