Please Help me with this question Let fx x2 and gx x2 x

Please Help me with this question

Let f(x) = x^2 and g(x) = x^2 - x + 1. Show that f is theta(g) using the definition (which means that you are not allowed to use a theorem for polynomial functions). For that you need to find constants A, B, c satisfying: A|g(x)| lessthanorequalto |f(x)| lessthanorequalto B|g(x)| for all x > c.

Solution

The only difference between the function f(x) and g(x) is of the factor 1-x, the value of f(x) will lead g(x), if the value of (1-x) is less than zero, which implies that x>1

Hence the value of constant c will be 1

For the first inequality we can write

A * |x^2 - x + 1| <= |x^2|

The given thing can hold for the value of A=1 given that x>1

Now we need to find the constraint for B

Bx^2 - Bx + B >= x^2

(B-1)x^2 - Bx + B >= 0

For the quadratic equation must always be greater than zero

Discriminant must be real

B^2 - 4B(B-1) >= 0

B(B - 4B + 4) >=0

B(4 - 3B)>=0

Hence the value of B will be 4/3