I tried researching and found that I can use a system of linear equations and solve by an inverse matrix to find the cubic equation given 4 points which satisfy the function f(x) of the general form f(x)=ax3+bx2+cx+dI can also find a cubic of the form ax3+d with no x2 or x term from 2 points, however I was wondering how one would go about finding a full general form cubic given only the minimum and maximum.Example minimums could be (-1,4) and (2,3)

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Randall Odom · Accepted Answer

Well, since the extrema are the roots of the derivate, it would be nice to have the minimum and the maximum (and an initial condition too.)

publilandiaik8 · Accepted Answer

First find a quadratic that has zeros at the x-values of your points. Then integrate. Then adjust the constant to get the correct y-values. Using your example, we need a quadratic that passes through the points (-1,0) and (2,0). So it must have the form g(x+1)(x−2)=gx2−gx−2g.  Integrate this to get f(x)=g3x3−g2x2−2gx+.  The y-values of your points insist that f(-1)=4 and f(2)=3. This gives two linear equations:  f(−1)=−g3−g2+2g+C=4  f(2)=8g3−2g−4g+C=3  which we can solve to get g=19 and C=9127. This gives  f(x)=127x3−118x2−29x+9127.

I tried researching and found that I can use a system of linear equations and solve by an inverse ma

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