Is 1 / ( x )^2 strictly greater than 1/(x) for all x?

2 Answers
Oct 22, 2015

No

Explanation:

1/x^2 <= 1/x if x>=1
color(white)("XXX")for example 1/(2^2) = 1/4 < 1/2

Oct 22, 2015

See explanation.

Explanation:

If x<0:

When x is negative, 1/x will be negative, but 1/x^2 will always be positive (because if x inRR, then x^2>=0). Therfore, 1/x^2 will be greater than 1/x.

If x=0:

Both expressions will be undefined. Therefore, you can't compare them.

If 0"<"x<1:

When x is between 0 and 1, 1/x^2 will be greater than 1/x.

If x=1:

Both expressions will be equal (they will both be 1).

If x>1:

When x is greater than 1, 1/x will be greater than 1/x^2. This is because in 1/x, you are dividing 1 by x only once, while in 1/x^2, you are dividing 1 by x twice.