Find a second degree polynomial P such that PO fO PO fO an

Find a second degree polynomial P such that P(O) = f(O), P?(O) = f?(O) and P?(O)=f?(O) for f(x)=4sinx-cos4x.

Solution

f(x)= 4sin x - cos 4x implies f(0) = -1

f\'(x) = 4 cos x + 4 sin 4x implies f\'(0) = 4

f\"(x)= -4 sin x + 16 cos 4x implies f\"(0) = 16

Now suppose,

P(x) = ax^2 + bx + c hence P(0) =c

P\'(x) = 2ax + b hence P\'(0)= b

P\"(x)= 2a hence P(0) = 2a

By comparing initial conditions,

c=-1

b=4

a=8

Hence P(x) = 8x^2 + 4x -1