Maximum Turning Point Calculator
Find critical points, extrema, and analyze function behavior with our advanced calculus tool
Function Definition
Analysis Options
Analysis Results
Critical Points
Enter a function to find its critical points.
Extrema Classification
Critical points will be classified as local maxima, minima, or saddle points.
Inflection Points
Points where the concavity changes will appear here.
Function Graph
📚 Calculus Reference
Critical Points
Points where f'(x) = 0 or is undefined. Potential locations of local maxima or minima.
First Derivative Test
If f'(x) changes from + to -, it's a local maximum. From - to + indicates a local minimum.
Second Derivative Test
If f''(x) > 0 at a critical point, it's a local minimum. If f''(x) < 0, it's a local maximum.
Inflection Points
Points where f''(x) = 0 and changes sign, indicating a change in concavity.
Graph Behavior
f'(x) > 0 ⇒ increasing, f'(x) < 0 ⇒ decreasing. f''(x) > 0 ⇒ concave up, f''(x) < 0 ⇒ concave down.
Common Derivatives
x^n → nx^(n-1), e^x → e^x, ln(x) → 1/x, sin(x) → cos(x), cos(x) → -sin(x)
🔢 Example Functions
Note: This calculator uses numerical methods to approximate derivatives and find roots. For exact symbolic solutions, consider using a computer algebra system. Results may vary slightly from exact analytical solutions.