In wave motion of a string both kinetic energy and potential energy are minimum at $y=y_\text{max}$ then why does the string come down again?
I've noticed that everyone is pretty bothered at the statement that both the kinetic energy & potential energy are minimum at the top. But though seems to be apparently-contradictory, it is actually true. In fact at the top, the string has zero kinetic energy as well as zero elastic potential energy.
So, I'm providing a bit context here:
In order to set up a wave on a stretched string, the driving force at the end of the string provides energy. This energy is not retained at the source; it flows along the string at the wave speed.
The string transports energy as both kinetic energy & elastic potential energy.
To send a sinusoidal wave along a previously straight string, the wave must stretch the string. As a string of length $dx$ oscillates transversely, its length must increase & decrease in a periodic way if the string element is to fit the sinusoidal form.
When the string element is at its $y = A$, its length is normal undisturbed value $dx$. However, when the element is rushing through its $y = 0$, it has maximum stretch & thus maximum elastic potential energy.
Thus the oscillating string element has both its maximum kinetic energy & maximum elastic potential energy simultaneously at $y = 0$.
Source: Principles of Physics; Extended 9th edition by Walker, Resnick, Halliday.
So, when the string is at the top, it has no kinetic energy; also it is not stretched as is evident from the pic above; so it has no elastic potential energy.
As everything in nature tries to attain the lowest energy possible, what brings that string element back to its original position?
You've not understood then what wave is. Wave transports energy without any net movement of any material-medium. Since, this part is at top now, it must have to come back so that there is no net displacement of the material-medium. This is quite a trivial (& poor) reasoning though.
But the main cause is that the source is continuously providing energy which is being spontaneously transported through the string. So, when the top-part of the string has no energy, the nearby-part of the string at the left which is now at the lowest position & thus has maximum energy exerts tension to the top-part & brings it downward in order to transfer the energy that is at left now to the right part. Thus the left part would now move at top since it is now lacking energy while the right-part that was previously at the top moves downward in the response of the tension & hence to receive the energy from the left & it would eventually again move upward by transferring this energy to the right. This is wave-motion.
Thus in a word, it is true that every system seeks to move towards having lower energy unless some external agent compels it to do otherwise. Here, though the top-part had no energy, but in order to transfer the energy from left-to-right, the left part of the string(as well as right) exerts tension-force on the top-part so it moves downward ultimately to gain the energy from the left & eventually it would move up again to transport this energy to the further right.