A twice-differentiable function satisfies the following conditions: • • • Throughout…

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

A twice-differentiable function

f

satisfies the following conditions: •

f(3)=5

•

f'(3)=-2

•

f''(3)=7

Throughout this question you will analyze a quadratic polynomial that locally models

f

near

x=3

. (1) Part 1: (a) Write the quadratic polynomial

P_2(x)

that agrees with

f

and its first two derivatives at

x=3

. (2) Part 2: (b) The value

f''(3)=7

is positive. Describe how this sign information affects the concavity of

P_2

and the quality of the approximation of

f

P_2

for

x>3

that are close to

3

. (3) Part 3: (c) Justify that any other quadratic polynomial that matches

f

x=3

together with the first two derivative values must be identical to the polynomial you wrote in part (a).