Where am I fault? Find g(x) when $\lim_{h\rightarrow 0}\frac{g(x+h)-2g(x)+g(x-h)}{h^2}=20x^3+6x.$ I got some problems here and I can't understand where am I fault. The task says:
if  $g$ is 2 times differentiable $,g'(0)=g(0)=1,$$$g''(x)=\lim_{h\rightarrow 0}\frac{g'(x)-g'(x-h)}{h},$$ and also $$20x^3+6x=\lim_{h\rightarrow 0}\frac{g(x+h)-2g(x)+g(x-h)}{h^2},$$ Prove that $$g(x)=x^5+x^3+x+1.$$
My Attempt:
$$\lim_{h\rightarrow 0}\frac{g(x+h)-2g(x)+g(x-h)}{h^2}$$
$$=\lim_{h\rightarrow 0}\frac{g(x+h)-g(x)+g(x-h)-g(x)}{h^2}$$
$$=\lim_{h\rightarrow 0}\frac{g(x+h)-g(x)}{h^2}+\frac{g(x-h)-g(x)}{h^2}$$
$$=\lim_{h\rightarrow 0}\frac{1}{2}\frac{g'(x+h)-g'(x)}{h}+\frac{1}{2}\frac{g'(x-h)-g'(x)}{-(-h)}$$
$$=\lim_{h\rightarrow 0}\frac{1}{2}\frac{g'(x+h)-g'(x)}{h}-\frac{1}{2}\frac{-(g'(x)-g'(x-h))}{-h}$$
$$=\lim_{h\rightarrow 0}\frac{1}{2}\frac{g'(x+h)-g'(x)}{h}-\frac{1}{2}\frac{g'(x)-g'(x-h)}{h}$$
$$=\frac{g''(x)}{2}-\frac{g''(x)}{2}=0.$$ Where I can't find $g(x).$ 

It is better to use L'Hospital's Rule here. We are given that $g$ is twice differentiable. So we have
\begin{align}
20x^{3} + 6x &= \lim_{h \to 0}\frac{g(x + h) - 2g(x) + g(x - h)}{h^{2}}\notag\\
&= \lim_{h \to 0}\frac{g'(x + h) - g'(x - h)}{2h}\text{ (via L'Hospital's Rule)}\notag\\
&= \frac{1}{2}\left(\lim_{h \to 0}\frac{g'(x + h) - g'(x)}{h} + \frac{g'(x - h) - g'(x)}{-h}\right)\notag\\
&= \frac{1}{2}\{g''(x) + g''(x)\}\notag\\
&= g''(x)\notag
\end{align}
Hence upon integrating the above equation we get $$g'(x) = 5x^{4} + 3x^{2} + A$$ for any constant $A$. Since $g'(0) = 1$ it means that $A = 1$. Further integration gives $$g(x) = x^{5} + x^{3} + x + B$$ for any constant $B$. Since $g(0) = 1$ it follows that $$g(x) = x^{5} + x^{3} + x + 1$$ The mistake in your attempt is in applying L'Hospital's Rule. Here the limit is taken as $h \to 0$ and hence the differentiation for L'Hospital Rule needs to be done with respect to $h$ (and not with respect to $x$ as you have done). Moreover as I have shown in my solution there is no need to split the $2g(x)$ term separately as $g(x) + g(x)$.
