Question

For the function $f\left(t\right)=\frac{e^t}{1+e^t},$

• A. $f\left(t\right)$ has a minimum at $t=\frac{1}{e}$
• B. $f\left(t\right)$ has a minimum at $t=1$
• C. $f\left(t\right)$ has a maximum at $t=e$
• D. $f\left(t\right)$ has no extreme value

Answer 1

The correct option is D $f\left(t\right)$ has no extreme value

Here $f\left(t\right)=1-\frac{1}{1+e^t}$
$\Rightarrow f^{\prime }\left(t\right)=\frac{e^t}{(1+e^t{)}^2}$
Now, $f^{\prime }\left(t\right)=0$
$\Rightarrow \frac{e^t}{(1+e^t{)}^2}=0$
But it has no real solution
$\Rightarrow f\left(t\right)$ has no extreme value.