Question

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

Answer 1

No

Explanation:

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

Answer 2

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.