minima and maxima of a voltage signal

Question

I wanted some help on how to approach on finding the minima and maxima of the following equation:

v(t)= $(t-2t^2)*e^{-0.5*|t|}$

Any help would be greatly appreciated.

Answer 1

Differentiating with respect to $ t $ and equating with 0, you have:

$$ (t-2t^2) . e^{-0.5 |t|} . (-0.5) \frac {t}{|t|} + e^{-0.5 |t|} . (1-4t) = 0$$

$$ => e^{-0.5 |t|} (-\frac {t}{2|t|} (t - 2t^2 ) +1 - 4t) = 0$$

Since there is no real solution of $ e^x = 0 $, so we can say: $$ -t^2 + 2t^3 + 2|t| - 8t|t| = 0$$

You solve the equation to get values of $ t $ and check if those values give maxima or minima value in the function by taking double derivative of $ t $ .