Show that the number of solutions of the equation $f(x) = 4x^5+x^3-2x-m$ has at least one solution and at most 3 solutions for any real values of $m$.
What I have done so far:
$f(x)=4x^5+x^3-2x-m\\f'(x)=20x^4+3x^2-2=(4x^2-1)(5x^2+2)\\f(m)=4m^5+m^3-3m=m(4m^4+m^2-3)=m(4m^2-3)(m^2+1)$
What I do next is to find test certain values of $m$
Case 1:
$-\frac 34<m\le0\\
f(-\frac 12)=\frac 34-m>0\\
f(-2)=-132-m<0\\
f(\frac 12)=-\frac34-m<0\\
f(2)=132-m>0$
Hence, by the Intermediate Value Theorem, I can conclude that there are at least 3 roots in Case 1. I will use Rolle's Theorem to prove that there are exactly 3 roots when $-\frac34<m\le0$
Case 2: $0<m<\frac34$
I will use similar argument to show that there are exactly 3 roots.
Case 3: $m\le-\frac{\sqrt3}{2}$
$f(-\frac 12)=\frac 34-m<0\\
f(m)<0\\
f(\frac 12)=-\frac34-m<0$
And yes, this is one of the problems I have encountered. I can sort of argue that $x=-\frac12$ is the local maximum and $x=\frac12$ is the local minimum and hence, there are no roots. I can even use Rolle's Theorem to show that there cannot be any roots. However, what value of $m$ can I substitute to show that in this particular case, there is at least one root by the Intermediate Value Theorem. (After doing this, I can use Rolle's Theorem to show that there is only 1 root). I will encounter the same problem for $m\ge\frac{\sqrt3}{2}$
But the greatest problem I have now is what values of $x$ do I substitute into $f(x)$ for $-\frac{\sqrt3}{2}<m\le-\frac34$ and $\frac34\le m<\frac{\sqrt3}{2}$ to show that there is at least 1 solution and at most 3 solutions?
Any help or any other solutions are appreciated! Thank you!
Best Answer
Accidentally deleted comment. Leaving as answer:
For any $m$ you've found that there are exactly $2$ real solutions to $f'=0$, namely $x=\pm \frac{1}{2}$. Rolle's theorem says that between any two solutions to $f=0$, there is a solution to $f'=0$, so that shows there are at most $3$ solutions.
There is at least one because the polynomial is odd degree. More generally, you can see that $\lim_{x\to -\infty}f(x)=-\infty$ and $\lim_{x\to \infty}f(x)=\infty$ so by the intermediate value theorem there has to be at least one solution.