To find the value of \(a \ge b\) such that the sum of the cubes of the roots of the quadratic equation \(x^2 - (a - 2)x + (a - 3) = 0\) is minimized, we start by understanding the problem using Vieta's formulas. Let's denote the roots of the equation by \(\alpha\) and \(\beta\).
The quadratic equation is given as:
\[ x^2 - (a - 2)x + (a - 3) = 0 \]
By Vieta's formulas, we know:
We need to find the sum of the cubes of the roots, which is \(\alpha^3 + \beta^3\). The formula for \(\alpha^3 + \beta^3\) in terms of \(\alpha + \beta\) and \(\alpha \beta\) is:
\[ \alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 - \alpha\beta + \beta^2) \]
Using the identity \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\), we find:
\[ \alpha^2 + \beta^2 = (a - 2)^2 - 2(a - 3) \]
\[ \alpha^2 + \beta^2 = a^2 - 4a + 4 - 2a + 6 = a^2 - 6a + 10 \]
Thus, \(\alpha^3 + \beta^3\) becomes:
\[ \alpha^3 + \beta^3 = (a - 2)((a^2 - 6a + 10) - (a - 3)) \]
\[ = (a - 2)(a^2 - 6a + 10 - a + 3) \]
\[ = (a - 2)(a^2 - 7a + 13) \]
To find the minimum value of \((a - 2)(a^2 - 7a + 13)\), we can examine the critical points by differentiating it with respect to \(a\) and setting it equal to zero if necessary.
However, we can bypass this by examining integer options provided:
The minimum is clearly at \(a = 3\), where the evaluated expression equals 1. Thus, the correct answer is 3.