Extrema

Q.

The minimum value $4e^{2x}+9e^{-2x}$ is

1. $11$

2. $12$

3. $10$

4. $14$

Q.

If $p\left(x\right)=4+3x-x^2+5x^3$ then, find $p(-1)$.

Q. Let $A=a_{ij}$ be a matrix of order $3,$ where
$a_{ij}=\{\begin{array}{cc}x;& \text{if}i=j,x\in R\\ 1;& \text{if}|i-j|=1\\ 0;& \text{otherwise}\end{array},$
then which of the following hold(s) good:

1. for $x=2,A$ is a diagonal matrix
2. $A$ is a symmetric matrix
3. Let $f\left(x\right)=detA$, then the function$f\left(x\right)$ has only maxima
4. Let $f\left(x\right)=detA$, then the function$f\left(x\right)$ has both the maxima and minima

Q. If $f\left(x\right)=x^2-2x;f:[0,3]\to A$, then $f$ is

1. not an injective function
2. a surjective function if $A=[-1,3]$
3. an injective function
4. a surjective function if $A=[0,3]$

Q. The maximum of $f\left(x\right)=\frac{\log x}{x^2}(x>0)$ occurs, when $x$ is equal to

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

Q. Let $f\left(x\right)=\frac{x^2+3}{[x+1]},0\le x\le 2$, where $[.]$ is the greatest integer function. Then the sum of the least value and the greatest value of $f\left(x\right)$ is

Q. List Match Sets
$\begin{array}{cccc}& \text{List - I}& & \text{List - II}\\ \text{I}& \text{The sides of a rectangle of greatest}& & \\ & \text{perimeter which is inscribed in a semicircle}& \text{(P)}& 108\\ & \text{of radius}\sqrt{5}\text{are}{\lambda }_1\text{and}{\lambda }_2\text{then}4({\lambda }_1^2+{\lambda }_2^2)\text{equals}& & \\ \text{II}& \text{If the number of points of minima of}f\left(x\right)=& \text{(Q)}& 2\\ & \left|\frac{x^2-2}{x^2-1}\right|\text{is}\lambda \text{then}18\lambda \text{is}& & \\ \text{III}& \text{If}f\left(x\right)=e^{2x}-2(a^2-21)e^x+8x+5& \text{(R)}& 68\\ & \text{is monotonically increasing for all}xϵR,& & \\ & \text{then the number of integers in range of 'a' are}& & \\ \text{IV}& \text{The volume of a rectangular closed box is 72}& & \\ & \text{and the base sides are in the ratio 1 : 2.}& \text{(S)}& 54\\ & \text{The least total surface area is}& & \end{array}$

Which of the following is the only CORRECT combination?

1. (III), (P)
2. (IV), (P)
3. (III), (Q)
4. (II), (P)

Q. If $x=-1$ and $x=2$ are extreme points of $f\left(x\right)=\alpha \log |x|+\beta x^2+x$ then :

1. $\alpha =-6,\beta =\frac{1}{2}$
2. $\alpha =-6,\beta =-\frac{1}{2}$
3. $\alpha =2,\beta =-\frac{1}{2}$
4. $\alpha =2,\beta =\frac{1}{2}$

Q. Let $f\left(x\right)$ be a differentiable function on $[0,8]$ such that $f\left(1\right)=6,f\left(2\right)=\frac{1}{3},f\left(3\right)=8,f\left(4\right)=-2,f\left(5\right)=5,f\left(6\right)=\frac{1}{5},\text{and}f\left(7\right)=-\frac{1}{3}$ If the minimum number of roots of the equation $f^{\prime }\left(x\right)-f^{\prime }\left(x\right)\left(f\right(x){)}^2=0\text{is}\lambda \text{then}\frac{\lambda }{11}$ is

Q. Let $f\left(x\right)$ be a differentiable function on $[0,8]$ such that $f\left(1\right)=6,f\left(2\right)=\frac{1}{3},f\left(3\right)=8,f\left(4\right)=-2,f\left(5\right)=5,f\left(6\right)=\frac{1}{5},\text{and}f\left(7\right)=-\frac{1}{3}$ If the minimum number of roots of the equation $f^{\prime }\left(x\right)-f^{\prime }\left(x\right)\left(f\right(x){)}^2=0\text{is}\lambda \text{then}\frac{\lambda }{11}$ is

Q.

If $ax+\frac{b}{x}\le$ c for all positive x, where a, b > 0, then

1. $ab<\frac{c^2}{4}$

2. $ab\ge \frac{c^2}{4}$

3. $ab\ge \frac{c}{4}$

4. $ab=c/4$