consider the function $f(x) = -2x^2 + 2x - 8$. $f(x)$ has a critical point at $x = a$.

find the value of $a$:
$a = $

at $x = a$, does $f(x)$ have a local min, a local max, or neither? type in your answer as lmin, lmax, or neither.

Question

consider the function $f(x) = -2x^2 + 2x - 8$. $f(x)$ has a critical point at $x = a$.

find the value of $a$:
$a = $

at $x = a$, does $f(x)$ have a local min, a local max, or neither? type in your answer as lmin, lmax, or neither.

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Critical Points", "First Derivative Test", "Local Extrema Classification" ], "new_concepts": [], "current_concepts": [ "Critical Points", "First Derivative Test", "Local Extrema Classification" ] } ## Find the derivative of the function $$ \begin{aligned} f(x) &= -2x^2 + 2x - 8 \ f'(x) &= -4x + 2 \end{aligned} $$ ## Solve for the critical point A $$ \begin{aligned} f'(x) &= 0 \ -4x + 2 &= 0 \ x &= 0.5 \ A &= 0.5 \end{aligned} $$ ## Classify the critical point using the second derivative $$ \begin{aligned} f''(x) &= -4 \ f''(0.5) &= -4 < 0 \end{aligned} $$ ## Question 1 Find the value of $A$: $A =$ 0.5 ## Question 2 At $x = A$, does $f(x)$ have a local min, a local max, or neither? Type in your answer as LMIN, LMAX, or NEITHER. LMAX { "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

Solve for the critical point A

Classify the critical point using the second derivative

Answer:

Question 1

Question 2