consider the following function:
\f(x) = \frac{1}{2}x^4 - 16x + 1\
step 3 of 4: determine where the function is concave up and concave down. enter your answers in interval notation.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Concavity", "Second Derivative Test" ], "current_concepts": [ "Concavity", "Second Derivative Test" ] } ## Find the first and second derivatives of the function $$ \begin{aligned} f(x) &= \frac{1}{2}x^4 - 16x + 1 \ f'(x) &= 2x^3 - 16 \ f''(x) &= 6x^2 \end{aligned} $$ ## Determine the sign of the second derivative $$ \begin{aligned} 6x^2 &\ge 0 \quad \text{for all } x \in \mathbb{R} \ f''(x) &> 0 \quad \text{for } x \in (-\infty, 0) \cup (0, \infty) \ f''(x) &= 0 \quad \text{at } x = 0 \end{aligned} $$ ## Identify intervals of concavity $$ \begin{aligned} \text{Concave up on: } & (-\infty, 0), (0, \infty) \ \text{Concave down on: } & \text{Never concave down} \end{aligned} $$ Concave up on: $(-\infty, 0), (0, \infty)$ Concave down on: Never concave down { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Concavity" ] }

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Explanation:

Find the first and second derivatives of the function

Determine the sign of the second derivative

Identify intervals of concavity

Answer: