The derivative is \( f'(x) = 9x^2 - 10x + 2 \).
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- Factor as \( (x - 3)^2 = 0 \).
- The root is \( x = 3 \) (double root).
- Find the derivative of \( f(x) = 3x^3 - 5x^2 + 2x - 7 \) and evaluate it at \( x = 2 \).
- Evaluate at \( x = 2 \): \( f'(2) = 9(2)^2 - 10(2) + 2 = 36 - 20 + 2 = 18 \).
- The integral \( \int (4x^3 - 2x + 1) \, dx \) is:
- So, the integral is \( x^4 - x^2 + x + C \).
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