Let's try a=1 and different values of c. Notice that the graph of the quadratic function y=x^2-c is a vertical translation of the graph of the parent function y=x^2 down c units. Similarly, the graph of y=- x^2+c is a vertical translation of the graph of y=- x^2 up c units.
Values a=1, c=1
Let's start with a=1 and c=1. We will determine whether the point (0,0) satisfies the inequalities.
Inequality
Substitution of ( 0, 0)
Result
Conclusion
y≥ x^2-1
0? ≥ 0^2-1
0≥ -1 ✓
Yes
y≤ - x^2+1
0? ≤- 0^2+1
0≤ 1 ✓
Yes
Let's graph these quadratic inequalities. Since they are not strict, the curves will be solid.
As we see above, for a=1 and c=1, the intersection of y≤ - ax^2+c and y≥ ax^2-c is not empty.
Values a=1, c=- 1
Let's change the direction of the translations and check a=1 and c=-1. We can again use the point (0,0) as a test point.
Inequality
Substitution of ( 0, 0)
Result
Conclusion
y≥ x^2+1
0? ≥ 0^2+1
0≱ 1 *
No
y≤ - x^2-1
0? ≤- 0^2-1
0≰ -1 *
No
Let's graph these inequalities.
We see that, for a=1 and c=-1, the intersection of y≤ - ax^2+c and y≥ ax^2-c is empty.
Conclusion
We have seen an example where the intersection is empty, and another where the intersection is not empty. Therefore, we can conclude that the given statement is sometimes true.
Extra
Can the intersection be empty for negative a?
It is interesting to note that for negative values of a the situation is different. In this case, the intersection cannot be empty. We will illustrate this for the case when a=-1 and c=-1.
