I'm working with some exercises about calculus and I'm stuck with one of them. I think that I'm near to solve it, but I don't know how to proceed. The exercise is
Let $f:[0,1]\to\mathbb{R}$ be a twice differentiable function over $[0,1]$ such that $f(0)=f'(0)=f'(1)=0$ and $f(1)=1$. Prove, using Taylor polinomials, that if $f\left(\frac{1}{2}\right)\geq \frac{1}{2}$ then there exist $x_1\in\left(0,\frac{1}{2} \right)$ such that $f''(x_1)\geq 4$ and if $f\left(\frac{1}{2} \right)\leq\frac{1}{2}$, then there exist $x_2\in \left(\frac{1}{2},1\right)$ such that $f''(x_2)\leq -4$.
First, we suposse that $f\left(\frac{1}{2}\right)\geq \frac{1}{2}$ (I think that the another case when $f\left(\frac{1}{2}\right)\leq \frac{1}{2}$ is analogous). By contradiction, suposse that for all $x\in\left(0,\frac{1}{2}\right)$ holds $f''(x)<4$. By the fact that $f$ is twice differentiable, we can take its first degree Taylor polonimial centered at $x=0$. Then $f(x)=P_{1,f,0}(x)+R_{1,f,0}(x)$ where $R_{1,f,0}(x)$ is the remainder at $x=0$. But by hypothesis $f(0)=f'(0)=0$. Thus $P_{1,f,0}(x)=0$ and then $f(x)=R_{1,f,0}(x)$. By the Lagrange theorem for the remainder, then $$f(1)=\dfrac{f''(c_1)}{2!}=1$$for some $c_1$ between $0$ and $1$. By the same argument $$f\left(\frac{1}{2} \right)=\dfrac{f''(c_2)}{2!}\geq \frac{1}{2}$$for some $c_2$ between $0$ and $1$. Thus $f''(c_2)\geq 1$. But, from here, I don't know how to derive a contradiction or how to going on. What can I do?
Taylor's expansion around $0$: $$f(x)=f(0)+xf^\prime(0)+\frac{x^2}2 f^{\prime\prime}(c_x)=\frac{x^2}2 f^{\prime\prime}(c_x)\tag{1}$$ for all $x\in [0,\frac 1 2]$ and some $c_x\in (0, x)\subset (0,\frac 1 2)$
Suppose that $f^{\prime\prime}(u)<4$ for all $u\in (0,\frac 1 2)$. Then, using $(1)$, we obtain that for all $x\in[0,\frac 1 2]$, $f(x) < \frac{\left(\frac 1 2\right)^2} 2 4 = \frac 1 2$ (remember that $c_x\in(0,\frac 1 2))$. Evaluate at $x=\frac 1 2$ and obtain $f(\frac 1 2)<\frac 1 2$.
For the other case: A similar reasoning should be applied by using a Taylor expansion around 1.