Mathematics - Application of Derivative Question with Solution | TestHub

Let $f(x)=x^5-5x+1$. Because $f(1)f(-1)<0$, there exists at least one root, say $\alpha$, of $f(x)=0$ in $(-1,1)$.

If there is another root $\beta$ such that $\alpha <\beta$ in $(-1,1)$, then by Rolle's Theorem applied to $[\alpha ,\beta ]$, it follows that there exists $\gamma \in (\alpha ,\beta )$ such that $f^{\prime }(\gamma )=0$. This means $5{\gamma }^4-5=0$, so ${\gamma }^4=1$, which implies $\gamma =\pm 1$. However, since $\gamma \in (\alpha ,\beta )\subset (-1,1)$, we must have $\gamma \ne \pm 1$, which is a contradiction. Hence, the number of roots of $f(x)=0$ in $(-1,1)$ is 1.