I am trying to graph a function with roots at $x=1$ with a multiplicity of $2$, $x=4$ with a multiplicity of $1$ that is $4$ degrees or higher. One end behavior is that as $x$ approaches infinity, $f(x)$ must approach negative infinity. The function must also be even.
Currently I have $f(x)=-(x-1)^2(x-4)$ but I am not sure how to modify this equation to include a local min at $x=3$. I am not allowed to use any form of calculus so derivatives are not allowed. What can I do to include a local min?
Here are your listed conditions:
root at $x=1$ with multiplicity $2$.
root at $x=4$ with multiplicity $1$.
degree at least $4$.
$\lim_{x \to \infty}=-\infty$
local minimum at $x=3$.
Even function
A polynomial that would satisfies this is
$$-(x+4)(x+3)^2(x+1)^2(x-1)^2(x-3)^2(x-4)$$