can a quartic graph have 2 turning points?

Question

Is it possible for a quartic equation when graphed to have 2 turning points and 1 point of inflection. If so what is an example of such equation? I am asking this because i have seen many quartic equations that have 3 or 1 turning point but never seen a quartic graph with 2 turning points.

Answer 1

You can tell the behavior of a polynomial $p(x)$ at $\pm\infty$ by looking at the term of highest degree, because for large $|x|$ the polynomial $$p(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots +c_0$$ is very close, in a relative sense, to $c_nx^n$.

This means that if $n$ is even we must have that the sign as $x\to\infty$ is the same as $x\to -\infty$ because this is true of $c_nx^n$. This can only happen if there are an odd number of turning points. Similarly, for odd $n$ the sign as $x\to\infty$ is the opposite of the sign as $x\to -\infty$, so there must be an even number of turning points.