Extra credit question

This is worth 5 points extra credit if done properly and turned in by May 4.

Get this as an ipynb file for IJulia.

Let $f$ be a 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. Here are some tools for our toolbelt:

using Plots
using Roots
Base.ctranspose(f::Function) = D(f)
tangent(f, c) = x -> f(c) + f'(c)*(x-c)

tangent (generic function with 1 method)

We can graph two tangent lines using this pattern:

f(x) = exp(x)
a,b = 1, 2
plot(f, 1, 2)
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 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 $b=2$.