Consider the function for

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

Problem

Consider the function

g(x)=5\sin x - x

for

0\le x\le 2\pi

. Exact values are expected unless otherwise indicated. (a) Find

g'(x)

and solve

g'(x)=0

for

x

(0,2\pi)

. (b) For each solution in part (a), determine whether it lies in the open interval

(0,2\pi)

and thus qualifies as a candidate for an absolute extremum. (c) Evaluate

g(x)

at every candidate from part (b) and at the endpoints

x=0

and

x=2\pi

. (d) Identify the absolute maximum and absolute minimum values of

g

[0,2\pi]

, including the corresponding

x

-coordinates. Provide exact values and show all reasoning.