Is there anything reliable known about who actually discovered the Chebyshev polynomials and what the motivation and circumstances were?

The reason why I am interested in knowing, is that I needed a solution for a variant of those polynomials: instead of all extrema having the same magnitude, I wanted to have them attain predefined values in a fixed order (I have found a solution for that problem, but involves a system of polynomial equations) and I wonder, whether the definition of the Chebyshev polynomials has been "guessed" or developed for a specific problem.

Edit:

at the request of @Hans, here is formal definition of my problem: given a sequence $(y_1,\ ...\ y_{n-1}), (y_{i+2}-y_{i+1})(y_{i+1}-y_i)<0$ of values, determine a polynomial $p(x)$ of degree $n$ and, $\ n$-$1$ abszissas $\ \xi_1 <,\ ...,\ <\xi_{n-1}$, so that $\ p(\xi_i)=y_i, p'(\xi_i)=0$

It should be noted that the polynomials that I am looking for, have no special properties, except for the predefined values in the extrema. The leading coefficient can be set to $1$ and the constant term to $0$.