If f(x)=frac{2^x}{2^x+sqrt{2}}, x in R, then | vedclass

Name: PDF Generator App | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

$f(x)=\frac{2^x}{2^x+\sqrt{2}}$
$f(x)+f(1-x)=\frac{2^x}{2^x+\sqrt{2}}+\frac{2^{1-x}}{2^{1-x}+\sqrt{2}}$
$=\frac{2^{x}}{2^{x}+\sqrt{2}}+\frac{2

Vedclass · Accepted Answer

$\frac{81}{2}$

Vedclass · Answer

$f(x)=\frac{2^x}{2^x+\sqrt{2}}$
$f(x)+f(1-x)=\frac{2^x}{2^x+\sqrt{2}}+\frac{2^{1-x}}{2^{1-x}+\sqrt{2}}$
$=\frac{2^{x}}{2^{x}+\sqrt{2}}+\frac{2}{2+\sqrt{2} 2^{x}}=\frac{2^{x}+\sqrt{2}}{2^{x}+\sqrt{2}}=1$
Now, $\sum_{k=1}^{81} f\left(\frac{k}{82}ight)=f\left(\frac{1}{82}ight)+f\left(\frac{2}{82}ight)+\ldots \ldots .+f\left(\frac{81}{82}ight)$
$=f\left(\frac{1}{82}ight)+f\left(\frac{1}{82}ight)+\ldots \ldots+f\left(1-\frac{2}{82}ight)+f\left(1-\frac{1}{82}ight)$
${\left[f\left(\frac{1}{82}ight)+f\left(1-\frac{1}{82}ight)ight]+\left[f\left(\frac{2}{82}ight)+f\left(1-\frac{2}{82}ight)ight]+\ldots .40 	ext { cases }+f\left(\frac{41}{82}ight)}$
$(1+1+\ldots . .+1) 40 	ext { times }+\frac{2^{1 / 2}}{2^{1 / 2}+2^{1 / 2}}$
$40+\frac{1}{2}=\frac{81}{2}$

If $f(x)=\frac{2^x}{2^x+\sqrt{2}}, x \in R$, then $\sum_{k=1}^{81} f\left(\frac{k}{82}\right)$ is equal to :

Similar Questions

The number of one-one function $f :\{ a , b , c , d \} \rightarrow$ $\{0,1,2, \ldots ., 10\}$ such that $2 f(a)-f(b)+3 f(c)+$ $f ( d )=0$ is

Let $A=\{0,1,2,3,4,5,6,7\} .$ Then the number of bijective functions $f: A \rightarrow A$such that $f(1)+f(2)=3-f(3)$ is equal to $.....$

Domain of the function $f(x) = {\sin ^{ - 1}}(1 + 3x + 2{x^2})$ is