Show that fx 0x2 by definition of BigOh find such constants

Show that f(x) = 0(x^2) \"by definition\" of Big-Oh (find such constants that necessary inequality holds). f(x) = x^2 + 5x - 3 Determine whether the following equalities are true or not. x^3 - x = 0(x^4) X^3 - x = 0(x^2)

Solution

1) Now here

|f(x)| = x² + 5x -3

<= |x²| + |5x| + |-3|

<= x² + 5x + 3 for all x>0

<= x² + 5x²+ 3x² for all x>1

<= 9x² for allx>1

We can conclude that f(x) = O(x^²) for constant C= 9 and k = 1 that holds inequality.

2)

With the same defined above we will check each option and prove LHS is equal to RHS or not.

a) x^³ - x

f(x) = x^³ - x

<= |x^³| + |-x|

<=  x^³ + x for x>0

<= x^³ + x^3 for all x>1

<= 2x^³

Here as per conclusion LHS and RHS are not equal

and same goes to option b