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Inflection Point Calculator

Find inflection points and analyze function behavior

Results

Points Found

Critical Point
x = -0.9500, y = 0.0000
Critical Point
x = -0.8500, y = 0.0000
Critical Point
x = -0.7500, y = 0.0000
Critical Point
x = -0.6500, y = 0.0000
Critical Point
x = -0.5500, y = 0.0000
Critical Point
x = -0.4500, y = 0.0000
Critical Point
x = -0.3500, y = 0.0000
Critical Point
x = -0.2500, y = 0.0000
Critical Point
x = -0.1500, y = 0.0000
Critical Point
x = -0.0500, y = 0.0000
Inflection Point
x = -0.9500, y = 0.0000
Inflection Point
x = -0.8500, y = 0.0000
Inflection Point
x = -0.7500, y = 0.0000
Inflection Point
x = -0.6500, y = 0.0000
Inflection Point
x = -0.5500, y = 0.0000
Inflection Point
x = -0.4500, y = 0.0000
Inflection Point
x = -0.3500, y = 0.0000
Inflection Point
x = -0.2500, y = 0.0000
Inflection Point
x = -0.1500, y = 0.0000
Inflection Point
x = -0.0500, y = 0.0000

Derivatives

First Derivative
3x^2
Second Derivative
2x^1

Solution Steps

First Derivative
3x^2
Find rate of change
Second Derivative
2x^1
Find concavity change
Critical Points
x = -0.95, -0.85, -0.75, -0.65, -0.55, -0.45, -0.35, -0.25, -0.15, -0.05
Points where first derivative equals zero
Inflection Points
x = -0.95, -0.85, -0.75, -0.65, -0.55, -0.45, -0.35, -0.25, -0.15, -0.05
Points where second derivative equals zero

Analysis

Analysis
Found 20 points of interest
Range
Range is appropriate for analysis
Verification
Cross-check results with graphical analysis
Applications
Use inflection points to analyze function behavior

Understanding Inflection Points

Definition

An inflection point is a point on a curve where the concavity changes from concave upward to concave downward, or vice versa. At these points, the second derivative changes sign.

Key Concepts

  • First derivative (f'(x)) shows rate of change
  • Second derivative (f''(x)) shows concavity
  • Inflection points occur where f''(x) = 0 and changes sign
  • Critical points occur where f'(x) = 0 or undefined

Applications

  • Optimization problems
  • Economic analysis (marginal cost/revenue)
  • Population growth models
  • Physical systems analysis