## Spring 2018, problem 60

Let $f,g:[a,b]\to [0,\infty)$ be two continuous and non-decreasing functions such that each $x\in [a,b]$ we have $$\int^x_a \sqrt { f(t) }\ dt \leq \int^x_a \sqrt { g(t) }\ dt \ \ \textrm{and}\ \int^b_a \sqrt {f(t)}\ dt = \int^b_a \sqrt { g(t)}\ dt.$$ Prove that $$\int^b_a \sqrt { 1+ f(t) }\ dt \geq \int^b_a \sqrt { 1 + g(t) }\ dt.$$