Q:

Impulse as a function of time - When should I differentiate it?

If impulse $I$ varies with time $t$ as $$I(\text{kg  m s}^{-1})= 20t^2-40t$$ The change in momentum is minimum at which time  
This is the question.
Impulse is change in momentum.
Here I am asked at what time the change in momentum will be zero?  
For this, why should I differentiate the above function?
$$\frac{dI}{dt} = 40t - 40$$ and $$40t - 40 = 0$$ because it is minimum
But why should not I equal it to zero without differentiating?

A:

The change in momentum is minimum at which time ... This is the question. ... Here i am asked at what time the momentum will be zero?

Asking "the change in momentum is minimum at which time" and "at what time the momentum will be zero" are two different questions.
You don't need to differentiate to find when the impulse is 0.
You should differentiate to find when the impulse reaches its minimum value.

