Let f be a function of a real variable into the real domain : f(x) = x^2 - 2*x + 1. Find the roots and the extremum of the function f.

Let f be a function of a real variable into the real domain : f(x) = x^2 - 2x + 1. Find the roots and the extremum of the function f.i) Root finding The function f is a 2nd degree polynomial. The root finding formula is hence applicable (Reminder if f(x) =ax^2+bx+c a 2nd degree polynomial, a, b and c real variables then its roots are defined by x = (-b +- sqrt(b^2-4ac))/(2a) ) The determinant of the 2nd degree polynomial is delta = (-2)^2-411 = 0 . The function hence only has one repeated root given by x = (-(-2)+sqrt(0))/21 = 1 . ii) Finding the extremum To find the extremum of a function we need to analyse the behaviour of its 1st derivative. f is continuous in the real domain, its derivative is hence defined for all real x . f'(x) = 2x - 2 f'(x) = 0 implies x = 1 . The extremum is hence located at x=1 and is the repeated root of the function. Before and after x=1, f(x) is strictly greater than 0, the extremum is hence a minimum. Those conclusions could have found by graphing the function.