Mathematics

Problem

For a family of functions

f_k(x)=kx^{2}+1

defined on the interval

[0,3]

, the numerical value

M

(where

M>1

) is prescribed to be the average value of

f_k

on that interval. a. (i) State the general formula for the average value of a continuous function on a closed interval. b. (ii) Using the definition, derive an explicit equation involving

k

and

M

that must be satisfied so that the average value of

f_k

[0,3]

equals

M

. c. (iii) Without solving the equation obtained in part (ii), explain whether

k

is directly proportional to

M

or inversely proportional to

M

. Justify your answer algebraically.

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