Let $a$ be an integer such that all real roots of the polynomial $2x^5+5x^4+10x^3+10x^2+10x+10$ lie in the interval $(a,a+1)$

Question

We need to find what $a$ is. Can I not solve this using the mean value theorem in the interval $(a,a+1)$?

I tried for a $c\in(a,a+1)$

$$f(c)=\int_a^{a+1}f(x)dx$$ but since all the roots lie in $(a,a+1), f(c)=0$ We could just integrate $f(x)$ and try different values of $a$ unless the equation is satisfied, but apparently this doesn't work.Why?

Answer 1

Let $f$ be the polynomial. Since $f$'s degree is odd, there is at least one root. This combined with the fact that the polynomial is strictly increasing, gives you that there is exactly one root. Now, we evaluate $f$ for some easy negative numbers (clearly the roots must be negative): $$f(-1)=3\text{ and }f(-2)=-34.$$ Therefore, the root(s) lies in $(-2,-1)$.