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Like so many things, the piano concerto was not so much invented as it evolved. A "concert piece" for a solo instrument set against a string (and sometimes woodwinds) accompaniment goes back as least as far as Vivaldi. Handel wrote a number of concertos for organ and small orchestra. Karl Phillip Emmanuel Bach was probably the earliest well-known composer to construct what is recognizably a piano concerto with "orchestra". Mozart and Haydn greatly developed it, and it reached maturity with Beethoven. It is basic to most composers since.
I doubt if there is any definitive answer to this, but there are a couple of contenders:
Later composers like Beethoven, Brahms, Mendelssohn, Sibelius, etc didn't write nearly so many each, but they tended to be quite a bit longer and very individual and expressive.
look it up on google, improvisation like a Classical Concerto Cadenza
The classical period ended about 1820, and the sax was not invented until after 1840, so no, the sax was not used in the classical period.
Symphony, solo concerto, solo sonata, string quartet, other chamber music genres.
The modern orchestra was first introduced during the classical period. Classical composers developed the genres of symphony and classical concerto (solo instrument and orchestra). Among the pioneer classical composers stand F. J. Haydn and W. A. Mozart.
symphony, concerto, sonata