Discussion
Let \(f(x) = e^{x}(2x^{2} + ax + 2 - a)\) . If \(f\) has a local minimum at x=2. the value of a is equal to
Answer: A
\(a =\) \(-9\).
\[ f'(x) = e^{x}[2x^{2} + (a + 4)x + 2] \]
A local minimum at 2 needs \(f'(2) = 0\): \(8 + 2(a + 4) + 2 = 0\), so \(a = -9\). Then \(f'(x) = e^{x}(2x^{2} - 5x + 2) = e^{x}(2x - 1)(x - 2)\), which changes from negative to positive at \(x = 2\), confirming a minimum.
The other values do not make \(f'(2) = 0\).
The other root of \(f'(x) = 0\), at \(x = \frac{1}{2}\), is a local maximum, since \(f'\) changes from positive to negative there.
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