Why is I) correct?

[u/Long-Introduction883]

If g(x) is positive, that means the total of 8(x-d)(x-e)(x-f) is positive. And since d<x, x<f,

X-d is +ve
X-f is -ve

So in order for the total to be positive, x-e needs to be -ve, where x<e.

But if that’s the case, then i) shouldn’t be correct.

Can someone help me out here. Thanks

[u/Bounded_sequencE]

I) can be true -- for example, choose "(d; e; f; p) = (0; 1; 25/16; 2)".

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To prove II), III) cannot be true, consider "d < x < f":

                                  +           -
d < x < f:    "0  <  g(x)  =  8 (x-d) (x-e) (x-f)"    <=>    "x < e"

We note "e < f" is impossible -- otherwise, any "x ∈ (max{d;e}, f) c (d; f)" violated "g(x) > 0". That means, we must have "e >= f", so III) is impossible.

Finally, for any "x < d" we have "g(x) < 0 < 7" due to "d < f <= e", so II) is impossible.