help with this math prob plz?

Question

Let f be a function that has derivatives of all the orders for all real numbers. Assume that f(3) =1, f '(3)=4, f "(3)=6, and f '"(3)=12

Does the linearixation of f underestimate or overestimate the values of f(x) near x=3?

Answer 1

Linearization:
f_l(x) ≈ f(a) + f'(a)(x-a)
= 1 + 4(x-3)= 4x -11

Taylor series, third-order:
f_t(x) ≈ f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + f'''(a)(x-a)³/3! + ε(x)
≈ 1 + 4(x-3) + 6(x-3)²/2! + 12(x-3)³/3!
= 1 + 4(x-3) + 3(x-3)² + 2(x-3)³

Then f_l(x) - f_t(x) = 3(x-3)² + 2(x-3)³
In the neighborhood of a = 3,
say x = a + h
f_l(3 + h) - f_t(3 + h) = 3(3 + h -3)² + 2(3 + h -3)³
f_l(3 + h) - f_t(3 + h) = 3h² + 2h³ = h²(3 + 2h)

So for small h i.e. x ≈ 3, (3+2h) will be positive, and h² is obviously always positive.

Hence linearization of f is an underestimate near x=3.