Consider the polynomial function and the point on its graph corresponding to .
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Consider the polynomial function f(x)=x3−6x2+9x+4 and the point C on its graph corresponding to x=1.
(i) Show that the instantaneous rate of change of f at x=1 is 0.
(ii) For a non–zero real number h, express the average rate of change of f over the interval [1,1+h] solely in terms of h.
(iii) Solve the equation obtained in (ii) when it equals the value found in (i) and hence determine every real number h=0 for which the average rate and instantaneous rate coincide.
(iv) Give a reasoned argument to show that no other real value of h satisfies the condition obtained in (iii).