A function satisfies for all real numbers

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

A function

h

satisfies

h'(x)=(x-3)(x+2)

for all real numbers

x

, and

h(0)=4

. (1) Identify all critical points of

h

and determine the open intervals on which

h

is increasing and the intervals on which it is decreasing. (2) Compute

h''(x)

and determine the intervals on which the graph of

h

is concave up and the intervals on which it is concave down. Find the

x

–coordinate of each inflection point. (3) Using the information from parts (a) and (b) together with the point

(0,4)

, sketch a possible graph of

y=h(x)

for

-4\le x\le 6

. Label all relative maxima, relative minima, and inflection points on your sketch.