Second derivative: \(f''(x) = 18x - 4\)
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- The function \(f(x) = 3x^3 - 2x^2 + x - 5\) has a critical point at \(x = 1\). Determine if it is a maximum, minimum, or neither.
- First derivative: \(f'(x) = 9x^2 - 4x + 1\)
- \(f'(1) = 9(1)^2 - 4(1) + 1 = 6\) (not zero, so not a critical point, but let's check second derivative)
- \(f''(1) = 18(1) - 4 = 14\) (positive, indicating a local minimum if \(f'(1) = 0\))
- Since \(f'(1)
- eq 0\), \(x = 1\) is neither a maximum nor a minimum.
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