Can a polynomial of degree $4$ have no turning points or no inflection points?

Question

Can a polynomial of degree $4$ have no turning points or no inflection points? If yes what is an example of polynomial of degree $4$ that show these feature. If no what is the minimum number of turning point or minimum number of points of inflection a degree $4$ polynomial can have.

Answer 1

A degree-4 polynomial is of the form $f(x)=ax^4+bx^3+cx^2+dx+e$ where $a\ne0$. Differentiating gives $f'(x)=4ax^3+3bx^2+2cx+d$. If $f(x)$ has no critical points, then $f'(x)$ must not be $0$ for any real $x$. Since $f'(x)$ is a cubic polynomial, this is not possible. Thus a quartic always has at least one critical point.

In general, a quartic polynomial must have at least one turning point (as do all polynomials of even degree) though it can have no point of inflection (e.g. the polynomial $f(x)=x^4$).