Roots of $x^4-3x^2+x-\sin x$

Question

Is there a logical/qualitative argument to find out the number of roots of $x^4-3x^2+x-\sin x$ ?

I tried plotting graphs for $x^4-3x^2+x$ and $\sin x$ to check the points of intersection but it only gets complicated.

Answer 1

Let $f(x)=x^4-3x^2+x-\sin x$. By inspection $f$ has a root at $0$, and we see that $f(-1)<0$, $f(1)<0$, $f(2)>0$, $f(-2)>0$, so, applying the Intermediate Value Theorem, $f$ has at least two other roots.

Consider now $f'(x)=4x^3-6x+1-\cos x$. Since $f'(0)=0$, $f$ has a double root at $0$, and so, counting multiplicities, $f$ has at least 4 roots.

Now use Rolle's Theorem. If $f$ had more than 4 roots, the fourth derivative of $f$ would have to have a root. (You fill in the details, starting with the fact that $f'$ would have to have at least 4 roots. Why?) But $f^{(iv)}(x)=24-\sin x\ge 23$. Thus, $f$ has precisely 4 roots.

Answer 2

Setting multiplicities aside, one approach would be to divide by $x$ and graph the even function

$$f(x)={\sin x\over x}-1$$

(noting that $f(0)=0$) and the odd function

$$g(x)=x^3-3x$$

Even a rough sketch shows the two curves cross transversally three times: once at $x=0$, once in the interval $[1,2]$, and once in the interval $[-2,-1]$, for a total of three roots. If you re-multiply by $x$, you can count an additional root at $x=0$.