I'm doing a maths problem in which a part of roller coaster track is missing. The gap must be filled by a polynomial equation and we know that it must smoothly connect with the points $(50,30)$ and $(100,0)$. At these points the gradient must be 0. The polynomials we have been doing are in the form $f(x)= A(x+a)(x+b)(x+c)$ etc.
So in summary I need an equation for a polynomial which goes exactly through $(50,30)$ and $(100,0)$
I think the equation must a cubic as we can then fill two of the parameters with $(x \pm100) $
Any help would be appreciated.
$\endgroup$ 11 Answer
$\begingroup$You're right if you want the function to have extreme points at $x=50$ and $x=100$, then it has to be at least of a third degree.
Now assume that $f(x) = Ax^3 + Bx^2 + Cx + D$. Now as $(50,30)$ and $(100,0)$ lie on the graph of $f$ we have that $30 = f(50)$ and $0 = f(100)$. Now take the derivative of $f$, which is $f'(x) = 3Ax^2 + 2Bx + C$. Now as $(50,30)$ and $(100,0)$ are extreme points we must have $f'(50) = 0$ and $f'(100) = 0$. This four equations should help you determine the 4 variables $A,B,C,D$ and finaly the polynomial $f$.
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