Mathematics - Application of Derivative Question with Solution | TestHub

Let the rectangle be $ABCD$.

$A=(t,0)$, $B=\left(t,e^{-t}\right)$, $C=\left(-t,e^{-t}\right)$, $D=(-t,0)$.

$ABCD$ $2t{e}^{-t}$ $f(t)=2t{e}^{-t}$.

To find the maximum area, we calculate the derivative of $f(t)$ with respect to $t$:

$$\frac{df}{dt}=2\left(t\left(-e^{-t}\right)+e^{-t}\right)=2e^{-t}(1-t)$$We analyze the sign of the derivative:

If $\frac{df}{dt}>0$, then $t\in (0,1)$.

If $\frac{df}{dt}<0$, then $t\in (1,\infty )$.

Thus, $t=1$ is the point of maxima.

The maximum area is $f(1)=\frac{2}{e}$.