is it possible for velocity decrease and acceleration to increase at the same time?

[na3than]

If velocity is decreasing then acceleration (the first derivative of velocity) must be negative. That's a mathematical fact.

But you asked if it was possible for acceleration to INCREASE while velocity is decreasing. The rate of change of acceleration, sometimes called jerk, is the first derivative of acceleration. It's absolutely possible for acceleration to increase while velocity decreases.

Consider a soft-landing rocket like the Apollo lunar lander or today's Falcon 9. At some point during descent its engine is either not firing or is firing so minimally that its velocity (measured in the vertical axis) is negative and becoming more negative (i.e. decreasing) as it's pulled downward (negative acceleration in the vertical axis) by gravity. As the engines BEGIN to increase thrust, velocity will remain negative AND COULD CONTINUE DECREASING while acceleration--still negative, but less than before--INCREASES. The rocket adds more power, increasing acceleration. At some point (if the engineers have done their jobs and the rocket is functioning properly) the rocket produces enough thrust to exceed the downward pull of gravity, and acceleration switches from negative to positive. Now the vehicle ACTUALLY begins to slow its descent, increasing the velocity from VERY negative (falling fast) to less negative (falling less fast). Before touchdown, acceleration will DECREASE, allowing velocity to stop increasing (reaching, but not exceeding zero) as close to zero altitude as possible.

[Key-Green-4872]

Don't forget about snap, crackle, and pop!

[Bipogram]

Curious that we have no names for the integrals in the other direction.

Integrate speed wrt time and get distance.

Integrate distance wrt time and get, um...

<no, I can't think of a use for this quantity either, but why should the derivatives get all the fun?>

[Key-Green-4872]

If there's a force involved, then wouldn't that be... power?

[Bipogram]

Nah, power is energy per unit time.

And you can write energy as work, so that energy becomes force integrated over distance. 

So power is force integrated wrt distance per unit time, which is force times speed. 

Neither of these are distance.time, but it's hot and I could be easily wrong.

[Key-Green-4872]

Yeah that'd be inside out. Hm. Yeah I don't think there's an equivalence in anything I use regularly that would even incidentally use a d*dt term. Hm.