What will be the nature of the equation sin(x + α)/sin(x + β)?

Category: QuestionsWhat will be the nature of the equation sin(x + α)/sin(x + β)?
Editor">Editor Staff asked 11 months ago

What will be the nature of the equation sin(x + α)/sin(x + β)?
 
(a) Possess only minimum value
 
(b) Possess only maximum value
 
(c) Does not possess a maximum or minimum value
 
(d) Data inadequate
 
The question was posed to me during an online exam.
 
The doubt is from Calculus Application topic in portion Application of Calculus of Mathematics – Class 12
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1 Answers
Editor">Editor Staff answered 11 months ago

Correct answer is (c) Does not possess a maximum or minimum value
 
The explanation: Let, y = sin(x + α)/sin(x + β)
 
Then,
 
dy/dx = [cos(x + α)sin(x + β) – sin(x + α)cos(x + β)]/sin^2(x + β)
 
= sin(x+β – x-α)/sin^2(x + β)
 
Or sin(β – α)/sin^2(x + β)
 
So, for minimum or maximum value of x we have,
 
dy/dx = 0
 
Or sin(β – α)/sin^2(x + β) = 0
 
Or sin(β – α) = 0  ……….(1)
 
Clearly, equation (1) is independent of x; hence, we cannot have a real value of x as root of equation (1).
 
Therefore, y has neither a maximum or minimum value.