Consider the polynomial function defined on the interval .

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

Problem

Consider the polynomial function

f(x)=x^{3}-6x^{2}+9x

defined on the interval

[0,5]

. (i) Factorise

f(x)

completely and hence find all its zeros that lie in the interval

[0,5]

. (ii) Using the sign of

f(x)

on each sub-interval determined by the zeros, identify on which portions of

[0,5]

the graph of

f(x)

lies above the

x

-axis and on which portions it lies below the

x

-axis. (iii) Hence calculate the total area enclosed between the curve

y=f(x)

and the

x

-axis from

x=0

x=5