Discussion
The function \(f(x) = x^{2} - x - 6\) is
Answer: A
The function has a minimum at x = 1/2.
Set the derivative to zero and check the second derivative:
\[ \begin{aligned} f'(x) &= 2x - 1 = 0 \Rightarrow x = \tfrac{1}{2} \\ f''(x) &= 2 \gt 0 \end{aligned} \]
A positive second derivative means a minimum; the parabola opens upwards because the coefficient of \(x^{2}\) is positive, so it has no maximum at all. The minimum value is \(f(1/2) = -6.25\). The options at \(x = -1/2\) come from a sign slip in solving \(2x - 1 = 0\).
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