Discussion
If \(f(x) = 2x^{3} - 3x^{2} - \lambda\ x + 1, x \in [0, 3]\) attains a local minimum at \(x = 2\) , then the value \(\lambda\) is equal to
Answer: C
\(\lambda\) is 12.
At a local minimum inside the interval, \(f'(2) = 0\).
\[ \begin{aligned} f'(x) &= 6x^{2} - 6x - \lambda \\ f'(2) &= 24 - 12 - \lambda = 0 \Rightarrow \lambda = 12 \end{aligned} \]
Check: \(f''(x) = 12x - 6\), and \(f''(2) = 18 \gt 0\), so x = 2 is a minimum. -12 comes from a sign slip with \(-\lambda\), and 36 from using \(f'(3)\). The other stationary point, from \(6x^{2} - 6x - 12 = 0\), is at x = -1, outside the interval [0, 3].
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