What are the critical values of f(x)=2x3−3x2? Select all that apply.

Question

BlackTom AI · Accepted Answer

To identify the critical values of f(x) = 2x^3 − 3x^2, we first differentiate the function to find f'(x).

Compute f'(x): f'(x) = 6x^2 − 6x = 6x(x − 1).

Critical values occur where the derivative is zero or undefined. Since f'(x) is a polynomial, it is defined for all real x, so we only consider the zeros of f'(x).

Set f'(x) = 0: 6x(x − 1) = 0, which gives x = 0 or x = 1.

Therefore, the x-values where the derivative equals zero are x = 0 and x = 1. These are the critical points arising from the condition f'(x) = 0. Since there are no points where the derivative is undefined, there are no other critical values to consider.

Option analysis:
- If a choice lists x = 0, this matches the root of f'(x) = 0 and aligns with the critical value condition.
- If a choice lists x = 1, this also matches the other root of f'(x) = 0 and aligns with the critical value condition.

In summary, the derivative must be zero at these x-values, and both x = 0 and x = 1 satisfy that requirement, while there are no other candidates because the derivative is defined for all real x and has only those two zeros.

What are the critical values of f(x)=2x3−3x2? Select all that apply. 多项选择题

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