Global Maxima

Q. The function $f\left(x\right)=\frac{x+1}{x^3+1}$ can be written as the sum of an even function $g\left(x\right)$ and an odd function $h\left(x\right)$. Then the value of $|g\left(0\right)|$ is

Q. The minimum value of $ata{n}^{2}x+bco{t}^{2}x$ equals the maximum value of a $si{n}^2\theta +bco{s}^2\theta wherea>b>0,$ then

1. a = b

2. a = 2b

3. a = 3b

4. a = 4b

Q. Let $S$ and $S^{\prime }$ be foci of an ellipse and $B$ be any one of the extremities of its minor axis. If $\Delta S^{\prime }BS$ is a right angled triangle with right angle at $B$ and area of $△S^{\prime }BS=8\text{sq. units}$, then the length of a latus rectum of the ellipse (in units) is :

1. $4\sqrt{2}$
2. $2$
3. $2\sqrt{2}$
4. $4$

Q. The minimum value of $ata{n}^{2}x+bco{t}^{2}x$ equals the maximum value of a $si{n}^2\theta +bco{s}^2\theta wherea>b>0,$ then

1. a = b

2. a = 2b

3. a = 3b

4. a = 4b

Q. The curve $f\left(x\right)=a^2{x}^3-0.5a{x}^2-2x-b$ has its local minima at $x=\frac{1}{3}$. If $f\left(\frac{1}{3}\right)>0,$ then

1. $a=-2$ and $b<-\frac{11}{27}$
2. $a=3$ and $b<-\frac{1}{2}$
3. $a=-2$ and $b>-\frac{1}{2}$
4. $a=3$ and $b<-\frac{3}{4}$

Q. If $\frac{dy}{dx}=\frac{y^3}{e^{2x}+y^2}$ and $y\left(0\right)=1,$ then

1. $y^2=e^{2x}-2e^{2x}\ln y$
2. $y^2=e^{2x}+2e^{2x}\ln y$
3. $y^2=e^{2x}+\frac{1}{2}{e}^{2x}\ln y$
4. $y^2=e^{2x}-\frac{1}{2}{e}^{2x}\ln y$