I am struggling to find the solution to this problem. If anyone could help to explain how to solve this problem to me, it would be really appreciated.
Let $$ f(x)=-\sqrt{3}x+(1+\sqrt{3}) $$ $$ g(x)= f(x)+C $$ When C varies, find the minimumn of the integral $$\int_0^2[g(x)]^2dx$$
The value of the integral is (bounds dropped for brevity): $$I(C)=\int(f(x)+C)^2dx=\int(f^2(x)+2Cf(x)+C^2)dx\\=\int f^2(x)dx+2C\int f(x)dx+C^2\int dx.$$
To find the extrema, find the roots of the first derivative with respect to $C$:
$$\frac{dI}{dC}=2\int f(x)dx+2C\int dx=0.$$
Solving this shouldn't be a big deal...
You will also easily check that the second derivative is positive.