The preliminary analysis


The given parabola is opened downward and has the vertex at x= -1 (given).

It means that the parabola has the maximum at x= -1.

On the coordinate plane, draw (mentally) such a parabola and horizontal line y = f(0).

It is clear, that this line is BELOW the vertex.

The point (0,f(0)) lies on the parabola and on the horizontal line.  
Hence, the symmetric point (-2, f(-2)) ALSO lies on the parabola and on the same horizontal line.

So, you conclude that at -2 < t < 0,  the points of parabola are ABOVE the points of the line.
Geometrically, the opposite is also OBVIOUS: if the point of the line is under the parabola, then  -2 < t < 0.

Also, notice that f(0) = f(-2) due to symmetry relative the axis x= -1.



      Now we are ready to analyze cases (I), (II) and (III).



So, if t is a number for which f(t) > f(0), it means that the parabola is above the line,
and hence, (I) is TRUE:  -2 < t < 0.  


Next, if t is a number for which f(t) > f(0), it means that the parabola is above the line,
and hence, (II) is NOT NOT.


And finally, if t is a number for which f(t) > f(0), it means that f(t) > f(0) > f(1),
i.e. (III) is TRUE.


    +--------------------------------------------------+
    |   At this point, the major part of the problem   |
    |                 is just done.                    |
    +--------------------------------------------------+


Having it, the rest is just   : 

       of 5 options (A), (B), (C), (D) and (E), only (C) is true.