7.1 inderfinite integral — Maths STD 12 Science — Question

$(S_1)$ there exists $\mathrm{x}_{1}, \mathrm{x}_{2} \in(2,4), \mathrm{x}_{1}<\mathrm{x}_{2}$, such that $f^{\prime}\left(x_{1}\right)=-1$ and $f^{\prime}\left(x_{2}\right)=0$

$(S_2)$ there exists $\mathrm{x}_{3}, \mathrm{x}_{4} \in(2,4), \mathrm{x}_{3}<\mathrm{x}_{4}$, such that $f$ is decreasing in $\left(2, x_{4}\right)$, increasing in $\left(x_{4}, 4\right)$ and $2 f^{\prime}\left(x_{3}\right)=\sqrt{3} f\left(x_{4}\right)$.

Then

$A_1=\left\{(x, y): x \geq 0, y \geq 0,2 x+2 y-x^2-y^2>1>x+y\right\}$

$A_2=\left\{(x, y): x \geq 0, y \geq 0, x+y>1>x^2+y^2\right\}$

$A_3=\left\{(x, y): x \geq 0, y \geq 0, x+y>1>x^3+y^3\right\}$

Denote by $\left|A_1\right|,\left|A_2\right|$ and $\left|A_3\right|$ the areas of the regions $A_1, A_2$ and $A_3$ respectively. Then,

1. $\text{y}_1=\frac{-1}{\text{a}+\text{b}\cos\text{x}}$
   
2. $\text{y}_2=\frac{\text{b}\sin\text{x}}{(\text{a}+\text{b}\cos\text{x})^2}$
   
3. $\text{y}_1=\frac{1}{\text{a}-\text{b}\cos\text{x}}$
   
4. $\text{y}_2=\frac{-\text{b}\sin\text{x}}{(\text{a}-\text{b}\cos\text{x})^2}$