Enter a polynomial, then explore its stationary points, sign changes, and nature using the graph and table.
f(x):
f'(x):
f''(x):
How to Use This Tool
1. Enter a polynomial
• Choose the degree (2–5).
• Fill in the coefficients for each power of \(x\).
• The tool automatically generates \(f(x)\), \(f'(x)\), and \(f''(x)\).
2. Understand the graph
• The blue curve shows the function \(f(x)\).
• Red points mark stationary points where \(f'(x)=0\).
• The graph auto‑scales to fit the curve and stationary points.
3. Read the combined sign + nature table
• Each interval row shows the sign of \(f'(x)\) and whether \(f(x)\) is increasing or decreasing.
• “Test x” is chosen automatically inside each interval.
• Rows labelled “At x = …” show the stationary points and their nature (max/min/possible inflection).
4. Interpreting the results
• A change from “+” to “–” indicates a maximum.
• A change from “–” to “+” indicates a minimum.
• No sign change indicates a stationary point of inflection.
Example: Analysing a cubic
With the default values, the tool uses:
\( f(x) = x^3 - 3x^2 - 4x + 12 \).
It finds two stationary points, one maximum and one minimum, and shows:
• where \(f'(x)\) is positive (increasing) or negative (decreasing),
• how the graph matches the sign and nature table.
