Optimization in mechanics

We have a curved plane which has a side view of a continous and double differential curve \(f:[0,1] \rightarrow [0,1],y=f(x)\) with \( y(0)=0 , y(1)=1\) and \(\dfrac{dy}{dx} \ge 0 , \dfrac{{d}^{2}y}{dx^{2}} \ge 0 \)

In this co-ordinate system \(1\) unit = \(1\) metre. Also gravity is present in the \(-y\) direction having a value of \(10 m/s^{2} \).

An point object is released from the point \((1,1)\) on the plane and it takes time \(t\) to reach the leave the plane from the bottom.

Let the minimum value of \(t\) be \(t_{min}\).

Find \( \lfloor 1000t_{min} \rfloor \)

The plane is smooth there is no friction present.You may use any computer software you want, any possible resources.

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