Avertical spring has a mass hanging from it, which is displaced from the equilibrium position and begins to oscillate. at what point does the system have the least potential energy?

the object has least potential energy at mean position of the SHM

Explanation:

If a block is connected with a spring and there is no resistive force on the system

In this case the total energy of the system is always conserved and it will change from one form to another form

So here we will say that

Kinetic energy + Potential energy = Total Mechanical energy

As we can say that total energy is conserved so here we have least potential energy when the system has maximum kinetic energy

So here we also know that at mean position of the SHM the system has maximum speed and hence maximum kinetic energy.

So the object has least potential energy at mean position of the SHM

see earlier answer

we have been given a quadratic function and we need to restrict the domain such that it becomes a one to one function.

we know that vertex of this quadratic function occurs at (5,2).

further, we know that range of this function is .

if we restrict the domain of this function to either or , it will become one to one function.

let us know find its inverse.

upon interchanging x and y, we get:

let us now solve this function for y.

hence, the inverse function would be if we restrict the domain of original function to and the inverse function would be if we restrict the domain to .