Question

Let $f\left(x\right)=a{x}^5+b{x}^4+c{x}^3+d{x}^2+ex$, where $a,b,c,d,eϵR$ and $f\left(x\right)=0$ has a positive root $\alpha$. Then,

• A. $f^{\prime }\left(x\right)=0$ has root ${\alpha }_1$ such that $0<{\alpha }_1<\alpha$
• B. $f^{\prime \prime }\left(x\right)=0$ has at least one real root
• C. $f^{\prime }\left(x\right)=0$ has at least two real roots
• D. all of the above

Answer 1

The correct option is D all of the above

From the given equation, we get $f\left(0\right)=0$
Hence, $f\left(x\right)$ has a real root ${\alpha }_0>0$.
$f\left(x\right)$ is continuous on $[0,{\alpha }_0]$ and differentiable on $(0,{\alpha }_0)$.
Hence there exists a ${\alpha }_1ϵ[0,{\alpha }_0]$ such that
$f^{\prime }\left({\alpha }_1\right)=0$ ...(Rolle's Theorem).
Consider $f^{\prime }\left(x\right)$.
It is continuous on $[0,{\alpha }_1]$ and differentiable on $(0,{\alpha }_1)$.
Hence, there exists a ${\alpha }_2ϵ[0,{\alpha }_2]$ such that
$f^{\prime }\left({\alpha }_2\right)=0$ ...(Rolle's Theorem).
And this can be repeated till $f^n=$ constant.
Hence, all of the above is true.