Problem

Find the derivative of the function \(f(x) = x^3 - 4x^2 + 6x - 2\), and then simplify the expression \(|f'(2)|\).

Answer

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Answer

Finally, we simplify the absolute value expression \(|f'(2)|\). Since \(f'(2) = 2\) and 2 is positive, we have \(|f'(2)| = |2| = 2\).

Steps

Step 1 ：First, we find the derivative of the function \(f(x) = x^3 - 4x^2 + 6x - 2\). Using the power rule for derivatives, we have \(f'(x) = 3x^2 - 8x + 6\).

Step 2 ：Then, we substitute \(x = 2\) into \(f'(x)\), giving us \(f'(2) = 3(2)^2 - 8(2) + 6 = 12 - 16 + 6 = 2\).

Step 3 ：Finally, we simplify the absolute value expression \(|f'(2)|\). Since \(f'(2) = 2\) and 2 is positive, we have \(|f'(2)| = |2| = 2\).

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