(i) can take both positive and negative values

A quadratic takes both positive and negative values if it has real roots, i.e., discriminant , and .

Here:

Step 1:

Step 2: Discriminant

Compute step by step:

Also, (otherwise it's not a quadratic)

So

(ii) For to be non-negative for all non-negative , we analyze two main conditions based on the graph's behavior:

Check boundary :

The value at must be non-negative: .

Case 1: No real roots ( ):

If the parabola never crosses the x -axis and with , it remains nonnegative everywhere.

Case 2: Real roots but not in the positive domain:

If roots exist, they must both be negative or zero so that for , the function remains .
For roots to be negative, the sum of roots must be negative and the product must be positive.

Sum < 0: .

Product or .

Combining these with gives the interval .

Combined Solution:

Joining and results in the final interval .

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