Math Problem Statement
Find the open intervals on which the function f(x)equalsaxsquaredplusbxplusc, anot equals0, is increasing and decreasing. Describe the reasoning behind your answer. Question content area bottom Part 1 When agreater than0, f is increasing on ▼ left parenthesis negative StartFraction b Over 2 a EndFraction comma infinity right parenthesis left parenthesis negative infinity comma negative StartFraction b Over 2 a EndFraction right parenthesis left parenthesis negative infinity comma StartFraction b Over 2 a EndFraction right parenthesis left parenthesis negative StartFraction b Over a EndFraction comma infinity right parenthesis left parenthesis StartFraction b Over a EndFraction comma infinity right parenthesis left parenthesis negative infinity comma StartFraction b Over a EndFraction right parenthesis because ▼ f double prime f prime is ▼ equal to 0 positive negative and decreasing on ▼ left parenthesis negative StartFraction b Over 2 a EndFraction comma infinity right parenthesis left parenthesis negative infinity comma negative StartFraction b Over 2 a EndFraction right parenthesis left parenthesis negative infinity comma StartFraction b Over a EndFraction right parenthesis left parenthesis negative infinity comma negative StartFraction b Over a EndFraction right parenthesis left parenthesis StartFraction b Over 2 a EndFraction comma infinity right parenthesis left parenthesis negative StartFraction b Over a EndFraction comma infinity right parenthesis because ▼ f double prime f prime is ▼ equal to 0. negative. positive. When aless than0, f is increasing on ▼ left parenthesis negative StartFraction b Over a EndFraction comma infinity right parenthesis left parenthesis negative StartFraction b Over 2 a EndFraction comma infinity right parenthesis left parenthesis negative infinity comma negative StartFraction b Over 2 a EndFraction right parenthesis left parenthesis StartFraction b Over a EndFraction comma infinity right parenthesis left parenthesis negative infinity comma negative StartFraction b Over a EndFraction right parenthesis left parenthesis negative infinity comma StartFraction b Over 2 a EndFraction right parenthesis because ▼ f double prime f prime is ▼ positive equal to 0 negative and decreasing on ▼ left parenthesis negative StartFraction b Over 2 a EndFraction comma infinity right parenthesis left parenthesis negative StartFraction b Over a EndFraction comma infinity right parenthesis left parenthesis negative infinity comma negative StartFraction b Over 2 a EndFraction right parenthesis left parenthesis StartFraction b Over a EndFraction comma infinity right parenthesis left parenthesis negative infinity comma negative StartFraction b Over a EndFraction right parenthesis left parenthesis negative infinity comma StartFraction b Over 2 a EndFraction right parenthesis because ▼ f prime f double prime is ▼ positive. negative. equal to 0.
Solution
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Functions
Calculus
Derivative
Formulas
f'(x) = 2ax + b (First Derivative of Quadratic Function)
Critical Point: x = -b / 2a
f'(x) > 0 (Increasing), f'(x) < 0 (Decreasing)
Theorems
First Derivative Test
Critical Points and Vertex of a Parabola
Behavior of Quadratic Functions (Increasing/Decreasing based on the sign of a)
Suitable Grade Level
Grades 10-12
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