A quadratic function is defined by p(x)=a x^ 2 +b x+c
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
A quadratic function p is defined by
p(x)=ax2+bx+c,
where a, b, and c are real constants.
The following information is known.
* p(3)=10.
* p(−1)=2.
* p′(3)=7.
* The inverse function p−1 exists in a neighborhood of x=3 and p(3)=10, and (p−1)′(10)=71.
Answer the following, showing all work.
(1) Using the given information, set up a system of equations involving a, b, and c. Clearly indicate the origin of each equation.
(2) Solve the system obtained in part (a) to determine the exact values of a, b, and c. Show each algebraic step.
(3) Verify that p is strictly monotonic on an open interval containing x=3, and hence justify the existence of p−1 on that interval.