For all $$x$$ in $$\left[ {0,1} \right]$$, let the second derivative $$f''(x)$$ of a function $$f(x)$$ exist and satisfy $$\left| {f''\left( x \right)} \right| < 1.$$ If $$f(0)=f(1)$$, then show that $$\left| {f\left( x \right)} \right| < 1$$ for all $$x$$ in $$\left[ {0,1} \right]$$.

Prove that the minimum value of $${{\left( {a + x} \right)\left( {b + x} \right)} \over {\left( {c + x} \right)}},$$
$$a,b > c,x > - c$$ is $${\left( {\sqrt {a - c} + \sqrt {b - c} } \right)^2}$$.