Mathematics - Application of Derivative Question with Solution | TestHub

Let $g(x)=(x-1)(x-2)(x-3)(x-4)$.

Let $f(x)=g(x)+c_0$, where $c_0=c-24$.

The equation $g(x)=0$ has four roots: $x=1,2,3,4$.

Also, $f^{\prime }(x)=g^{\prime }(x)$ and $f^{\prime \prime }(x)=g^{\prime \prime }(x)$.

By Rolle's theorem, $g^{\prime }(x)=0$ has at least one root in each of the intervals $(1,2)$, $(2,3)$, and $(3,4)$.

By Rolle's theorem, between any two roots of $f^{\prime }(x)=0$, $f^{\prime \prime }(x)=0$ has at least one root.