Let be the quadratic function defined for all real

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

Let

f

be the quadratic function

f(x)=mx^{2}+nx

defined for all real

x

, where

m

and

n

are real constants. Two design specifications must be met: 1. The average value of

f

on the interval

[-1,2]

6

. 2. The average value of

f

on the interval

[0,2]

7

. Answer the following. (1) Formulate a system of two equations in the unknowns

m

and

n

that encodes the two average value conditions. (2) Solve the system obtained in part (a) to determine the exact values of

m

and

n

. (3) Using your values of

m

and

n

, compute the average value of

f

[\! -2,\,2]

. Show your work.