Question

If `{(x-5)(x^2-2x+1)}/{(x-7)(x^2+2x+3)}` is positive for all real value of x,show that x has no value between 5 and 7?

Answer 1

Hence proved.

Explanation:

Let

`y=((x-5)(x^2-2x+1))/((x-7)(x^2+2x+3))`

Simplifying,

`y=((x-5)(x-1)^2)/((x-7)((x+1)^2+1))`

Now, note that the `(x-1)^2` and `(x+1)^2+1` are positive for all values of x

Thus, the sign of `y` depends only on the `(x-5)/(x-7)` term

For values between 5 and 7, The above term is -ve.

Thus,
For `y` to be +ve, `x` must not lie between 5 and 7.

Note: My answer is in no way be fit to be presented in an exam paper, it is only to help you understand the logic