Global Minima

Q.

A function $f\left(x\right)$ is given by $f\left(x\right)=\frac{5^x}{5^x+5}$, then the sum of the series: $f\left(\frac{1}{20}\right)+f\left(\frac{2}{20}\right)+f\left(\frac{3}{20}\right)+......+f\left(\frac{39}{20}\right)$ is equal to:

1. $\frac{19}{2}$

2. $\frac{49}{2}$

3. $\frac{39}{2}$

4. $\frac{29}{2}$

Q.

If the functions $f,g$ and $h$ are defined from the set of real numbers $\mathrm{ℝ}\to \mathrm{ℝ}$ such that $f\left(x\right)=x^2-1$, $g\left(x\right)=\sqrt{x^2+1}$ and $h\left(x\right)=\left\{\begin{array}{ll}0,& \mathrm{if}x<0\\ x,& \mathrm{if}x\ge 0\end{array}\right.$ then $h\circ \left(f\circ g\right)\left(x\right)$ is defined by

1. $x$

2. $x^2$

3. $0$

4. None of these.

Q. $f\left(x\right)=\frac{(x-2)(x-1)}{(x-3)}\forall x>3$. The minimum value of f(x) is equal to

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

Q. Which of the following statements is (are) TRUE?

1. $\underset{x\to 0}{lim}\frac{|2x-1|-|2x+1|}{x}$ does not exist.
   
2. The function $f\left(x\right)=\{\begin{array}{cc}2-x,& \text{if}x\le 1\\ x^2-3x+2,& \text{if}x>1\end{array}$ is differentiable at $x=1.$
3. The absolute difference between the maximum and the minimum value of the function $f\left(x\right)=3{\sin }^{4}x-{\cos }^{6}x$ is $4$.
4. The function $f\left(x\right)=x^5+10x^3+20x-18$ is strictly increasing $\forall x\in R$