May be not to differentiate it

Using Cauchy-Schwartz Inequality, $100= (x+(2-x))((50-x)+x) \ge \left(\sqrt{x(50-x)}+\sqrt{x(2-x)}\right)^2$ . Hence, $\sqrt{x(50-x)}+\sqrt{x(2-x)} \le 10$ . The equality holds if and only if $\frac{x}{50-x}=\frac{2-x}{x}$ , which means that $x=\frac{100}{52}=\frac{25}{13}$ .

Now, $c=\frac{25}{13}$ and $L=10$ an so $13c+L=25+10=35$ .

Differentiate the function, and set it equal to zero. We can trivially see that it's max occurs at x = 25/13, so c = 25/13 and f(c) = L = 10. Thus 25 + 10 = 35, trivial.