Taking It To 100

Each extrema (or "turning point") of a polynomial represents a place where the tangent line would have a slope of 0.

$f(x)= ax^2 + bx + c$ has a maximum of 1 extrema.

$f(x)=ax^3 + bx^2 + cx + d$ has a maximum of 2 extrema.

$f(x)=ax^4 + bx^3 + cx^2 + dx + e$ has a maxmimum of 3 extrema. (Example shown below.)

In general, the maximum number of extrema of a polynomial of degree $n$ is $n -1 .$ So a degree 100 polynomial will have at most $100 - 1 = 99$ places where the tangent line will have a slope of 0.

Let $P(x)$ be this polynomial. The degree of $P(x)$ is equal to 100.

We want to find the maximum number of solution to $P'(x)=0$ where $P'(x)$ is the derivative of $P(x)$ .

The degree of $P'(x)$ is equal to 99 therefore there are at most 99 solutions to $P'(x)=0$ .