Question

The response of a network is $i(t) = K t e^{-\alpha t}$ for $t \ge 0$, where $\alpha$ is real and positive. The value of $t$ at which $i(t)$ will become maximum is,

• A. $\alpha$
• B. $2\alpha$
• C. $1/\alpha$
• D. $\alpha^2$

Hint:

For functions of the form $t e^{-\alpha t}$, the maximum always occurs at $t = 1/\alpha$.

Answer 1

The Correct Option is C

Step 1: Write the given current function.
\[ i(t) = K t e^{-\alpha t} \]
Step 2: Differentiate $i(t)$ with respect to time.
\[ \frac{di(t)}{dt} = K \left( e^{-\alpha t} - \alpha t e^{-\alpha t} \right) \]
Step 3: Set the derivative equal to zero for maximum value.
\[ e^{-\alpha t} (1 - \alpha t) = 0 \]
Since $e^{-\alpha t} \neq 0$,
\[ 1 - \alpha t = 0 \]
Step 4: Solve for $t$.
\[ t = \frac{1}{\alpha} \]
Step 5: Conclusion.
The current $i(t)$ attains its maximum value at
\[ \boxed{t = \frac{1}{\alpha}} \]