Given this data table consists of 10 points:
x | -3 1 2 4 5 6 9 11 12 14
y | 2 3 -2 0 -3 1 -1 0 -2 1
Construct a polynomial P(x) such that its graph passes through the points in the data table and P(x) has a local maximum at M3(2,−2), a local minimum at M5(5,−3), and a saddle point at M8(11, 0).
For this problem, I first thought about Lagrange Polynomial to construct the function P(x):
(1807093x^9/61751289600)−(5058059x^8/3087564480)+(5106197x^7/137837700)−(13208401247x^6/30875644800)+(14385778741x^5/5613753600)−(191204226071x^4/30875644800)−(142786904483x^3/15437822400)+(23555534161x^2/285885600)−(8112434009*x/53603550)+(1318323/15470)
Here's the Desmos link for its graph: https://www.desmos.com/calculator/j2b9vofetd
But I can't think of any way to add Local Maximum/Minimum to the function, I tried to Derivative to get P'(x) then force the P'(x=minimum) = 0 but that would mess up the whole function. I even tried to add a "running" point but that doesn't work either.
Any idea about solving the issue would be greatly appreciated.