First slide

Maxima and minima

Question

The function $\frac{\sin (x + a)}{\sin (x + b)}$ ,has no maxima or minima if

Moderate

A

$b - a = nπ, n \in I$
B

$b - a = (2 n + 1) π, n \in I$
C

$b - a = 2 nπ, n \in I$
D

none of these

Solution

$f (x) = \frac{\sin (x + a)}{\sin (x + b)}$
$f^{'} (x) = \frac{\sin (x + b) \cos (x + a) - \sin (x + a) \cos (x + b)}{\sin^{2} (x + b)}$
$= \frac{\sin (b - a)}{\sin^{2} (x + b)}$
If $\sin (b - a) = 0, then f^{'} (x) = 0 or f (x)$ will be a constant ,i.e., $b - a = nπ or b - a = (2 n + 1) π or b - a = 2 nπ .$
Then f(x) has no minima

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