Local Maxima

Q.

The magnitude and amplitude of $\frac{(1+i\sqrt{3})(2+2i)}{\left(\sqrt{3}-i\right)}$ are respectively

1. $2,\frac{3\mathrm{\pi }}{4}$

2. $2\sqrt{2},\frac{3\mathrm{\pi }}{4}$

3. $2\sqrt{2},\frac{\mathrm{\pi }}{4}$

4. $2\sqrt{2},\frac{\mathrm{\pi }}{2}$

Q.

Find the value of $0.5737373....$ in fraction

1. $\frac{284}{497}$

2. $\frac{284}{495}$

3. $\frac{568}{990}$

4. $\frac{567}{990}$

Q. If $f\left(x\right)=\frac{\tan (x+\frac{\pi }{6})}{\tan x}$ attains local minimum at $x=a\pi$ in the interval $(\frac{\pi }{3},\pi )$ and the local minimum value is $b,$ then the value of $a+b$ is

1. $0$
2. $\frac{7}{2}$
3. $1$
4. $3$

Q. A differentiable function $f$ satisfies the relation $f\left(\frac{x+1}{x-1}\right)=2f\left(x\right)+\frac{3}{x-1}\forall x\in R-\left\{1\right\}.$ If
$\left(i\right)$ the range of the function excludes the interval $(a_1,a_2)$ on the real number line,
$\left(ii\right)$ relative maximum and minimum values of $f\left(x\right)$ occur at $x=b_1$ and $x=b_2,$
then the value of $a_1^2+a_2^2+b_1^2+b_2^2$ is

Q. If $f\left(x\right)=(\begin{array}{c}3x^2+12x-1,-1\le x\le 2\\ 37-x,2<x\le 3\end{array}$, then

1. f(x) is increasing in [-1, 2]
2. f(x) is continuous in [-1, 3]
3. f '(2) does not exist
4. f(x) has a maximum value at x = 2

Q.

Let $f\left(x\right)=\{\begin{array}{cc}|x^2-3x|+a,0\le x<\frac{3}{2}& \\ -2x+3x\ge \frac{3}{2}& \end{array}$ If f(x) has a local maximum at x =.

1. $a\le 0$

2. $a\le -\frac{9}{4}$

3. $a\ge -\frac{9}{4}$

4. $a=3$

Q. The number of integral values of $a$ for which the function $f\left(x\right)=\{\begin{array}{cc}{x}^3+1,& x\le 0\\ a^2-5a+7+x& x>0\end{array}$
has maxima at $x=0$ are

1. \N
2. 1
3. 2
4. infinite

Q. The maximum value of the function $f\left(x\right)=3x^3-18x^2+27x-40$ on the set $S=\{x\in R:x^2+30\le 11x\}$ is :

1. $122$
2. $-122$
3. $222$
4. $-222$

Q.

Let $f:[a,b]\to R$ be a function such that for
$c\epsilon (a,b),f^1\left(c\right)=f^{11}\left(c\right)=f^{111}\left(c\right)=f^{tv}\left(c\right)=f^v\left(c\right)=0$ then

1. f has local extremum at x=c

2. f has neither local maximum nor local minimum at x = c

3. f is necessarily a constant function

4. It is difficult to say whether (a) or (b)

Q. Let $f\left(x\right)=\{\begin{array}{cc}\frac{b^3+b-2b^2-2}{b^2+5b+6}-x^2;& 0\le x<1\\ 3x-4;& 1\le x\le 3\end{array}$
where $b\in R$. If $f\left(x\right)$ has minimum value at $x=1,$ then the least integral value of $b$ is

Q. Let $S$ be the set which contains all possible values of $l,m,n,p,q,r$ for which
$A=\left[\begin{array}{ccc}{l}^2-3& p& 0\\ 0& m^2-8& q\\ r& 0& n^2-15\end{array}\right]$ be a non singular idempotent matrix. Then the sum of all the elements of the set $S$ is