Let $f(x)$ and $g(x)$ are two differential functions such that $f(x) = \frac{x^3}{2} + 1 - x \int_0^x{g(t)}dt $ and $g(x) = x - \int_0^1{f(t)}dt $. Also suppose $\alpha = min |f(x) - g(x) |$ for all values of $ x \in \mathbb R , \beta = $ minimum distance between the curves $f(x)$ and $g(x)$.
Two questions asked based on this given fact is:
1) Find the value of $\left( \alpha + \beta \right)$ $$(a) \frac{7\sqrt{2}}{3} \\(b) \frac{14}{3} \\(c) \frac{7}{3} \left( 1 + \frac{1}{\sqrt{2}} \right) \\(d) \frac{7}{3} \left( 1 + \sqrt{2} \right) $$
2) If $h(x) = \frac{f(x)}{g(x)}$ and $m$ = Number of points of extrema of $h(x)$ and $n$ = Number of critical points on the graph $y = h(x)$ , then $(m+n)$ is equal to $$(a) 3 \\(b) 4 \\(c) 5 \\(d) 6 $$
I could not get any idea to start solving the problem. Any suggestion will be highly appreciated.