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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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.