Let f: [0,1] - R satisfy f(xy) = f(x)f(y) for all x, y in [0,1] with f(0) not equal to 0. If y = y(x) solves dy/dx = f(x) with y(0) = 1, find the value of y(1/4) + y(3/4).

Question

Let f: [0,1] -> R satisfy f(xy) = f(x)f(y) for all x, y in [0,1] with f(0) not equal to 0. If y = y(x) solves dy/dx = f(x) with y(0) = 1, find the value of y(1/4) + y(3/4).

Accepted Answer

3. The functional equation forces f(x) = 1, so dy/dx = 1 and y = x + 1; therefore y(1/4) + y(3/4) = (5/4) + (7/4) = 3.

Let f: [0,1] -> R satisfy f(xy) = f(x)f(y) for all x, y in [0,1] with f(0) not equal to 0. If y = y(x) solves dy/dx = f(x) with y(0) = 1, find the value of y(1/4) + y(3/4).

Solution

Related JEE Advanced Maths questions