Suppose $f$ is three time differentiable function on $-1,1$ such that $f(-1)=0,f(1)=1$ and $f^{\text{'}}(0)=0.$ Now using Taylor theorem prove that $f^{\text{'}\text{'}\text{'}}(c)3$ for some cin$(-1,1).$ Hint: Divide the interval into two parts $-1,0$ and $0,1$ and use Taylor's thorem to obtain $f(c_1)+f(c_2)=6,$ where $c_{1}in(-1,0)$ and $c_{2}in(0,1).$ This implies one of these values greater than or equal to $3$