Mathematics

Problem

Let

h(x)=ax^{3}-6x+4

for real

a

. Consider the interval

[1,4]

. a. Compute the average rate of change of

h

[1,4]

and write your answer as a function of

a

. Show all algebraic steps. b. Find the instantaneous rate of change of

h

x=1

in terms of

a

. Clearly indicate any derivative you compute. c. Determine all real values of

a

for which the instantaneous rate of change found in part (b) equals the average rate of change found in part (a). Justify your solution algebraically. d. For each value of

a

obtained in part (c), discuss whether there must exist at least one

x

in the open interval

(1,4)

at which the instantaneous rate of change of

h

equals that same common rate. Provide a clear justification based on properties of the function on the interval.

Browse the problem library