The maximum value of the function \( f(x) = (x - 1)(x - 2)(x - 3) \) in the domain [0, 3] occurs at \( x = \) \_\_\_\_\_\_ (rounded off to two decimal places).
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Solution and Explanation
Now, evaluate \( f(x) \) at the critical points \( x = 2.58 \), \( x = 1.41 \), and at the endpoints \( x = 0 \) and \( x = 3 \).
\( f(0) = (0 - 1)(0 - 2)(0 - 3) = (-1)(-2)(-3) = -6 \)
\( f(3) = (3 - 1)(3 - 2)(3 - 3) = (2)(1)(0) = 0 \)
\( f(1.41) = (1.41 - 1)(1.41 - 2)(1.41 - 3) = (0.41)(-0.59)(-1.59) \approx 0.384 \)
\( f(2.58) = (2.58 - 1)(2.58 - 2)(2.58 - 3) = (1.58)(0.58)(-0.42) \approx -0.384 \)
Step 4: Conclusion
The maximum value occurs at \( x = 1.41 \), and the value of the function is approximately \( 0.384 \), but rounded to two decimal places, the maximum occurs at: \[ \boxed{1.41} \]
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