1. Given function

$f(x)=\frac{1}{e^x+2e^{-x}}, \qquad x\in \mathbb R$

We must examine both statements.

2. Analyze Statement-2 first

We need to check whether

$0<f(x)\le \frac{1}{2\sqrt 2}\quad \text{for all }x\in \mathbb R.$

Since $e^x>0$ and $e^{-x}>0$ for all $x$, we have

$e^x+2e^{-x}>0$

so

$f(x)=\frac{1}{e^x+2e^{-x}}>0.$

Now find the maximum value of $f(x)$.

Since $f(x)$ is the reciprocal of $e^x+2e^{-x}$, maximizing $f(x)$ is equivalent to minimizing

$g(x)=e^x+2e^{-x}.$

Differentiate:

$g'(x)=e^x-2e^{-x}.$

Set $g'(x)=0$:

$e^x-2e^{-x}=0$ $e^x=2e^{-x}$ $e^{2x}=2$ $x=\frac{1}{2}\ln 2.$

Now

$g''(x)=e^x+2e^{-x}>0,$

so this gives a minimum of $g(x)$.

Evaluate $g$ there:

Let $e^x=\sqrt 2$, then $e^{-x}=\frac{1}{\sqrt 2}$.

Hence

$g_{\min}=\sqrt 2+2\cdot \frac{1}{\sqrt 2}=\sqrt 2+\sqrt 2=2\sqrt 2.$

Therefore,

$f(x)\le \frac{1}{2\sqrt 2}.$

Combining with positivity,

$0<f(x)\le \frac{1}{2\sqrt 2}\quad \forall x\in \mathbb R.$

So Statement-2 is true.

3. Now check Statement-1

Statement-1 says:

$f(c)=\frac13 \quad \text{for some } c\in \mathbb R.$

From Statement-2, we know

$0<f(x)\le \frac{1}{2\sqrt 2}.$

Now compare:

$\frac{1}{2\sqrt 2}=\frac{1}{2.828\dots}>\frac13=\frac{1}{3}=0.333\dots$

Indeed,

$2\sqrt 2<3 \implies \frac{1}{2\sqrt 2}>\frac13.$

Also, $f$ is continuous on $\mathbb R$.

Now check some value:

$f(0)=\frac{1}{1+2}=\frac13.$

So directly, for $c=0$,

$f(c)=f(0)=\frac13.$

Hence Statement-1 is true.

4. Is Statement-2 the correct explanation for Statement-1?

Statement-2 tells us that the range satisfies

$0<f(x)\le \frac{1}{2\sqrt 2}.$

Since

$\frac13 < \frac{1}{2\sqrt 2},$

and $f$ is continuous, while also

$f(x)\to 0 \quad \text{as } x\to \pm \infty,$

the value $\frac13$ is indeed attained (in fact, directly at $x=0$).

More simply, Statement-2 establishes that the function takes positive values up to at least $\frac13$, and continuity ensures intermediate values are attained. So it serves as the intended explanation for Statement-1.

Thus both statements are true, and Statement-2 is the correct explanation of Statement-1.

5. Option check

• A: Both true, but not explanation — incorrect
• B: Statement-2 false — incorrect
• C: Statement-1 false — incorrect
• D: Both true, and Statement-2 is correct explanation — correct

Final Answer

$\boxed{\text{D}}$