Let be defined on the closed interval .

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

Let

h(x)=e^{\sin x}-\sin x

be defined on the closed interval

[0,2\pi]

. (a) Derive an explicit expression for

h'(x)

and determine all critical points of

h

in the open interval

(0,2\pi)

. (b) For each subinterval of

(0,2\pi)

determined by the critical points from part (a), establish the sign of

h'(x)

. Support each conclusion with appropriate analysis. (c) Classify every interior critical point as a relative maximum or relative minimum, and state the corresponding coordinates

(x,h(x))

. (d) Evaluate

h

x=0

and

x=2\pi

. Using these values together with the classifications from part (c), identify the absolute maximum and absolute minimum values of

h

[0,2\pi]

, and specify where they occur.

Related theory

Increasing and Decreasing Functions
Curve Analysis
Maxima and minima

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