the function $f(x) = 2x^3 - 24x^2 + 42x - 1$ has two critical numbers.

the smaller one is x =

and the larger one is x =

Question

the function $f(x) = 2x^3 - 24x^2 + 42x - 1$ has two critical numbers.

the smaller one is x =

and the larger one is x =

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Critical Points" ], "new_concepts": [], "current_concepts": [ "Critical Points", "Derivative Equations" ] } ## Find the derivative of the function $$ \begin{aligned} f(x) &= 2x^3 - 24x^2 + 42x - 1 \ f'(x) &= 6x^2 - 48x + 42 \end{aligned} $$ ## Set the derivative to zero and solve for x $$ \begin{aligned} 6x^2 - 48x + 42 &= 0 \ 6(x^2 - 8x + 7) &= 0 \ 6(x - 1)(x - 7) &= 0 \ x = 1 \quad &\text{or} \quad x = 7 \end{aligned} $$ ## Identify the smaller and larger critical numbers $$ \begin{aligned} x_{\text{smaller}} &= 1 \ x_{\text{larger}} &= 7 \end{aligned} $$ The function $f(x) = 2x^3 - 24x^2 + 42x - 1$ has two critical numbers. The smaller one is x = 1 and the larger one is x = 7 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Critical Points" ] }

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QUESTION IMAGE

Question

Explanation:

Find the derivative of the function

Set the derivative to zero and solve for x

Identify the smaller and larger critical numbers

Answer: