\(g ( - x ) = x^3 - 5 x - 2\).
If we find the least integral upper bound \(c\) on the set of zeros of \(g ( - x )\), then \(- c\) will be the greatest integral lower bound on the set of negative zeros of \(g ( x )\).
| \(1\) | \(0\) | \(-5\) | \(-2\) | |
| \(1\) | \(1\) | \(1\) | \(-4\) | \(-6\) |
| \(2\) | \(1\) | \(2\) | \(-1\) | \(-4\) |
| \(3\) | \(1\) | \(3\) | \(4\) | \(10\) |
The first entry to give a row of all non-negative numbers is \(3\), thus \(-3\) is the greatest integral lower bound on the set of zeros of \(g ( x )\).
This is the same answer as problem 2.5.8. In problem 2.5.8 we had \(f(x)=x^3-5x+2\) and in problem 2.5.9 we had \(g(x)=-x^3+5x-2\) which is the negative of \(f(x)\). A function and it's negative have the same zeros.