Differentiation of trigonometric functions and differentiation rules — Question 37

Original QCE Vault practice · 5 marks

Q37 · Practice questionTechnology-freeComplex familiar5 marks

QUESTION 37 (5 marks)

For

h(x)=\sin x\cos x

0\le x\le\pi

, determine the stationary points and classify them.

WORKED SOLUTION

5 marks

ANSWER

Maximum at

(\pi/4,1/2)

and minimum at

(3\pi/4,-1/2)

Worked solution

h'(x)=\cos^2x-\sin^2x=\cos2x

. Hence

\cos2x=0

gives

x=\pi/4,3\pi/4

. Evaluating

h

gives

1/2

and

-1/2

. The derivative sign changes

+\to-

\pi/4

and

-\to+

3\pi/4

Differentiates to

h'(x)=\cos2x

[1 mark]

Solves

\cos2x=0

on the interval.

[1 mark]

Determines both stationary-point coordinates.

[1 mark]

Classifies

\pi/4

as a local maximum.

[1 mark]

Classifies

3\pi/4

as a local minimum.

[1 mark]

Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.

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