find the interval in which the function is increasing or decreasing f(x) = (4sinx - 2x - xcosx ) / - Maths - Application of Derivatives

Question

find the interval in which the function is increasing or
decreasing f(x) = (4sinx - 2x - xcosx ) / (2+ cosx)

Duke Biswass · Accepted Answer

$\frac{4 \sin x - 2 x - x \cos x}{2 + \cos x}$

derivative of this function = $\frac{(4 \cos x - 2 - \cos x + x \sin x) (2 + \cos x) - (- \sin x) (4 \sin x - 2 x - x \cos x)}{{(2 + \cos x)}^{2}}$

= $\frac{\sin^{2} x + 4 \cos x - 1}{{(2 + \cos x)}^{2}}$

The function is increasing in the intervals where the derivative is positive and decreasing in the intervals where the derivative is negative.

In the derivative term the denominator is a square so it is always positive. So we only need to check the numerator for sign.

For positive derivative $\sin^{2} x + 4 \cos x - 1 > 0$

⇒ $1 - \cos^{2} x + 4 \cos x - 1 > 0$

⇒ $\cos x (4 - \cos x) > 0$

⇒ $4 > \cos x$ and $\cos x > 0$

Now cosx has maximum value 1 so 4 would always be greater than cosx

we have to tae the other value cosx>0

⇒ the interval in which the given function is positive is $(- \frac{π}{2}, \frac{π}{2})$ or $(\frac{3 π}{2}, \frac{π}{2})$ as cosx is positive in this interval

similarly for the function to be decreasing $\cos x (4 - \cos x) < 0$

4 can never be less than cos x so we have to take values where cosx<0 which gives us $(\frac{π}{2}, \frac{3 π}{2})$

Hence the given function is increasing in the interval $(\frac{3 π}{2}, \frac{π}{2})$ and decreasing in the interval $(\frac{π}{2}, \frac{3 π}{2})$

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find the interval in which the function is increasing or decreasing.f(x) = (4sinx - 2x - xcosx ) / (2+ cosx)