Mathematics

Problem

Let

f(x)=ax^{2}+bx+c

with real numbers

a\ne 0,\,b,\,c

. For two distinct real numbers

p

and

q

with

p<q

, answer the following. a. (i) Show that the instantaneous rate of change of

f

at any

x

f'(x)=2ax+b

. b. (ii) Prove that the average rate of change of

f

[p,q]

a(p+q)+b

. c. (iii) Hence establish that there is exactly one point

c

(p,q)

where the instantaneous rate of change equals the average rate of change, and show that

c=\dfrac{p+q}{2}

. Justify the uniqueness of this point.