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Question: The function <img src="https://cdn.pureessence.tech/canvas_240.png?top_left_x=313&top_left_y=300&wid...

The function has no maximum or minimum if

(k an integer) –

A

b – a = kp

B

b – a¹ kp

C

b – a = 2kp

D

) None

Answer

b – a¹ kp

Explanation

Solution

Let ƒ(x) = , so that

ƒ¢(x) = sin⁡(x+β)cos⁡(x+α)−cos⁡(x+β)sin⁡(x+α)sin⁡2(x+β)\frac { \sin ( x + \beta ) \cos ( x + \alpha ) - \cos ( x + \beta ) \sin ( x + \alpha ) } { \sin ^ { 2 } ( x + \beta ) }

= sin⁡(α−β)sin⁡2(x+β)\frac { \sin ( \alpha - \beta ) } { \sin ^ { 2 } ( x + \beta ) }

Thus, ƒ(x) has no maximum or minimum if ­ƒ¢(x) ¹ 0. i.e. if sin (b – a) ¹ 0

Equivalently b – a ¹ kp, k Ī I.