Let $$g\left( x \right) = \log f\left( x \right)$$ where $$f(x)$$ is twice differentible positive function on $$\left( {0,\infty } \right)$$ such that $$f(x+1)=x f(x)$$. Then, for $$N=1, 2, 3, ..........$$
$$g''\left( {N + {1 \over 2}} \right) - g''\left( {{1 \over 2}} \right) = $$

Let $$\,\,\,$$$$f\left( x \right) = 2 + \cos x$$ for all real $$X$$.

STATEMENT - 1: for eachreal $$t$$, there exists a point $$c$$ in $$\left[ {t,t + \pi } \right]$$ such that $$f'\left( c \right) = 0$$ because
STATEMENT - 2: $$f\left( t \right) = f\left( {t + 2\pi } \right)$$ for each real $$t$$.

$${{{d^2}x} \over {d{y^2}}}$$ equals

If $$f(x)$$ is a twice differentiable function and given that $$f\left( 1 \right) = 1;f\left( 2 \right) = 4,f\left( 3 \right) = 9$$, then