This is worth 5 points extra credit if done properly turned in by the last class.
Let $f$ be a continuously differentiable function defined on the real line. M. Stemlund proved in 2001 (The College Mathematics Journal, 32, 194-196) the following:
The function $f(x)$ is a quadratic polynomial if and only if for any $a$ and $b$ the intersection point of the tangents lines to $f(x)$ at $(a, f(a))$ and $(b, f(b))$ happens at $x=(a+b)/2$.
We wish to investigate this. We will use the tangent function provided by the MTH229 package:
using MTH229
We can graph two tangent lines using this pattern:
f(x) = exp(x) a,b = 1, 2 plot(f, a, b) plot!(tangent(f, a)) plot!(tangent(f, b))
Here we can see the intersection point is not at (a+b)/2 = 1.5, but a bit to the right of that.
repeat the above for the function $f(x) = x^2$ and show graphically that the intersection point of the tangent lines is at $(a+b)/2$.
Pick another pair of values for a and b using a,b = sort(4*rand(2)) and show graphically that this relationship is not a coincidence.
Now, show numerically (using fzero or fzeros) that the intersection point is equal to (a+b)/2 for the function $f(x) = x^2 + 2x + 3$ with $a=1$ and $b=2$.