Find Lagrange's interpolating polynomial of degree two from the values below (2,0.69315)(2.5,0.91629)(3,1.09861) hence find f(2.7)?

To find Lagrange's interpolating polynomial of degree two, we first need to calculate the coefficients of the polynomial using the given points (2,0.69315), (2.5,0.91629), and (3,1.09861).

The Lagrange interpolating polynomial of degree two is given by:
P(x) = L0(x)f(x0) + L1(x)f(x1) + L2(x)f(x2)

where Li(x) = Π (x - xj) / (xi - xj) for j ≠ i

Calculating the coefficients:
L0(x) = (x - 2.5)(x - 3) / (2 - 2.5)(2 - 3) = (x - 2.5)(x - 3) / 0.5
L1(x) = (x - 2)(x - 3) / (2.5 - 2)(2.5 - 3) = (x - 2)(x - 3) / 0.25
L2(x) = (x - 2)(x - 2.5) / (3 - 2)(3 - 2.5) = (x - 2)(x - 2.5) / 0.5

Now, substitute the values of f(x0), f(x1), and f(x2) into the formula:
P(x) = L0(x) * 0.69315 + L1(x) * 0.91629 + L2(x) * 1.09861

Now, we can calculate the value of P(2.7) by substituting x = 2.7 into the polynomial P(x) we found above.