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.
a. Part 1: (a) Write the quadratic polynomial P2(x) that agrees with f and its first two derivatives at x=3.
b. Part 2: (b) The value f′′(3)=7 is positive. Describe how this sign information affects the concavity of P2 and the quality of the approximation of f by P2 for x>3 that are close to 3.
c. Part 3: (c) Justify that any other quadratic polynomial that matches f at x=3 together with the first two derivative values must be identical to the polynomial you wrote in part (a).