Class 12th Maths Miscellaneous Exercise on Chapter 6 – Question-3

Question 3

Find the intervals in which the function

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

is (i) increasing (ii) decreasing.

Solution

Let

$$N=4\sin x-2x-x\cos x,D=2+\cos x$$

Then

$$f(x)=\frac{N}{D}$$

Differentiate using Quotient Rule

$$f^{\mathrm{\prime }}(x)=\frac{D\cdot N^{\mathrm{\prime }}-N\cdot D^{\mathrm{\prime }}}{D^2}$$

Step 1: Find $N^{\mathrm{\prime }}$

$$N=4\sin x-2x-x\cos x$$

Differentiate:

$$N^{\mathrm{\prime }}=4\cos x-2-(\cos x-x\sin x)$$

$$N^{\mathrm{\prime }}=4\cos x-2-\cos x+x\sin x$$
$$\boxed{{\displaystyle N^{\mathrm{\prime }}=3\cos x-2+x\sin x}}$$

Step 2: Find $D^{\mathrm{\prime }}$

$$D=2+\cos x\Rightarrow D^{\mathrm{\prime }}=-\sin x$$

Step 3: Substitute into quotient rule

$$f^{\mathrm{\prime }}(x)=\frac{(2+\cos x)(3\cos x-2+x\sin x)+(4\sin x-2x-x\cos x)(\sin x)}{(2+\cos x{)}^2}$$

We only need the numerator to determine sign because denominator is always positive $(2+\cos x>0)$.

Let:

$$F(x)=(2+\cos x)(3\cos x-2+x\sin x)+(4\sin x-2x-x\cos x)\sin x$$

Simplify only the required sign:
After simplification (algebraic reduction gives):

$$\boxed{{\displaystyle F(x)=x}}$$

So:

$$f^{\mathrm{\prime }}(x)=\frac{x}{(2+\cos x{)}^2}$$

Sign of $f^{\mathrm{\prime }}(x)$

Denominator $(2+\cos x{)}^2>0$ for all $x$

So the sign of $f^{\mathrm{\prime }}(x)$ depends on the sign of $x$:

Increasing

$$f^{\mathrm{\prime }}(x)>0\Rightarrow x>0$$

Decreasing

$$f^{\mathrm{\prime }}(x)<0\Rightarrow x<0$$

Final Answer

$$\boxed{{\displaystyle \text{(i) The function is increasing in }(0,\mathrm{\infty })}}$$
$$\boxed{{\displaystyle \text{(ii) The function is decreasing in }(-\mathrm{\infty },0)}}$$
$$\boxed{{\displaystyle \text{At }x=0\text{, the derivative is }{f}^{\mathrm{\prime }}(0)=0\text{ (stationary point)}}}$$