= Solution
<Rolle theorem> states that if $h$ is continuous on the closed interval between two distinct points, differentiable between them, and has equal endpoint values, then $h'$ vanishes at an interior point.
Let $P_N(t)=\sum_{j=0}^Nf^{(j)}(0)t^j/j!$ and, for $x\ne0$, put
$$
M=\frac{f(x)-P_N(x)}{x^{N+1}},
\qquad h(t)=f(t)-P_N(t)-Mt^{N+1}.
$$
Then $h(x)=0$ and $h^{(j)}(0)=0$ for $0\leq j\leq N$. Applying Rolle's theorem successively $N+1$ times gives a point $\xi=\theta x$, $0<\theta<1$, where $h^{(N+1)}(\xi)=0$. Hence
$$
M=\frac{f^{(N+1)}(\theta x)}{(N+1)!},
$$
which is the stated <Taylor theorem with Lagrange remainder>.
If $f'=0$, the $N=0$ case gives $f(x)=f(0)+xf'(\theta x)=f(0)$. Thus $f$ is constant; equivalently this is the immediate corollary of the <mean value theorem>.
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