Let $f(x)=x^2+2x+2$,$g(x)=-x^2+2x-1$,and $a, b$ be the extreme values of $f(x)$ and $g(x)$ respectively. If $c$ is the extreme value of $\frac{f}{g}(x)$ (for $x \neq 1$),then $a+2b+5c+4=$

Let $f : R \to R$ be a function defined by $f(x) = \frac{4^x}{4^x + 2}$. What is the value of $f(\frac{1}{4}) + 2 f(\frac{1}{2}) + f(\frac{3}{4})$?

Let $R$ denote the set of all real numbers. Let $f: R \rightarrow R$ be a function such that $f(x) > 0$ for all $x \in R$,and $f(x+y)=f(x) f(y)$ for all $x, y \in R$. Let the real numbers $a_1, a_2, \ldots, a_{50}$ be in an arithmetic progression. If $f(a_{31})=64 f(a_{25})$,and $\sum_{i=1}^{50} f(a_i)=3(2^{25}+1)$,then the value of $\sum_{i=6}^{30} f(a_i)$ is:

The set of all $a \in R$ for which the equation $x|x-1|+|x+2|+a=0$ has exactly one real root is:

Let $f(x) = |x|$ and $g(x) = |x| + a$,where $a > 0$. For $0 \leq x \leq b$,the set $\{(x, y) \mid g(x) \leq y \leq f(x)\}$ represents all the points in the interior of:

If a function $g(x)$ is defined in $[-1, 1]$ and two vertices of an equilateral triangle are $(0, 0)$ and $(x, g(x))$ and its area is $\frac{\sqrt{3}}{4}$,then $g(x)$ equals :-