Relative Extrema Calculator
Find all relative maxima, minima, and saddle points with detailed step-by-step solutions using the second derivative test
Calculator Inputs
Enter your function and optional domain
Use ^ for powers (e.g., x^2, x^3). Example: x^3 - 6*x^2 + 9*x + 1
Quick Guide
- •Enter polynomial functions using ^ for powers
- •Example: x^3 - 3*x^2 + 2
- •Domain is optional (defaults to all real numbers)
- •Uses second derivative test for classification
How to Use This Calculator
Step-by-step guide to get accurate results
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What Are Relative Extrema?
Relative extrema are local peaks and valleys of a function's graph. A Local Maximum occurs where f(c) ≥ f(x) for all x near c (downward curve ∩). A Local Minimum occurs where f(c) ≤ f(x) for all x near c (upward curve ∪). Unlike absolute extrema (global high/low), relative extrema depend on nearby values only.
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How the Calculator Works
The calculator uses a systematic 3-step process to find and classify relative extrema.
Step 1: First Derivative
The calculator computes f'(x) = d/dx f(x). Setting f'(x) = 0 gives critical points where extrema may occur.
Step 2: Critical Points
Critical points are values of x where f'(x) = 0 or f'(x) is undefined. These are all candidate points for relative maxima or minima.
Step 3: Classification Tests
Second Derivative Test: f''(c) > 0 means Local Minimum, f''(c) < 0 means Local Maximum, f''(c) = 0 is Inconclusive. First Derivative Test: Check the sign change of f'(x) around each critical point.
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Core Formulas
First Derivative: f'(x) = d/dx f(x). Second Derivative: f''(x) = d²/dx² f(x). These formulas are the foundation for finding and classifying extrema.
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Detailed Example
Function: f(x) = x³ - 3x² + 2. First Derivative: f'(x) = 3x² - 6x = 3x(x-2), giving critical points x = 0, 2. Second Derivative: f''(x) = 6x - 6. At x=0: f''(0) = -6 < 0, so Local Max. At x=2: f''(2) = 6 > 0, so Local Min. Values: f(0) = 2, f(2) = -6. Result: Max(0, 2), Min(2, -6). Interactive Graph shows peak at x=0, valley at x=2, with derivative overlay.
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Why Use This Calculator?
Instant Results
Solve even complex functions in seconds without manual calculations.
Step-by-Step Solutions
Full derivative calculations explained clearly for learning.
Interactive Graphs
Zoomable plots with marked extrema for visual understanding.
100% Free
No paywalls or subscriptions required, unlimited use.
Accurate & Error-Free
Symbolic and numerical solving included for precision.
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Supported Functions
Polynomial Functions
x^n + ... for any degree polynomial.
Trigonometric Functions
sin(x), cos(x), tan(x) with periodic extrema detection.
Exponential Functions
e^x and other exponential expressions.
Logarithmic Functions
ln(x) and log functions.
Piecewise Functions
Functions defined in multiple pieces.
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Relative vs Absolute Extrema
Understanding the difference between local and global extrema.
Relative Extrema
Scope: Local neighborhood. Number: Multiple possible. Test: f'(x) = 0. Example: f(x) = x³ has min at x=0.
Absolute Extrema
Scope: Entire domain. Number: One max, one min. Test: f'(x) = 0 plus endpoints. Example: Check boundaries as well.
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How to Use This Calculator
Using the calculator is quick and straightforward. Just enter your function and get instant results.
Step 1: Enter Function
Enter your function, e.g., x^3-3x^2+2 or sin(x)+x.
Step 2: Set Interval (Optional)
Optionally set an interval [a,b] to restrict the domain.
Step 3: Calculate
Click Calculate to see points, derivative tests, graph, and export PDF. Trig Example: f(x) = x - 2sin(x) gives critical points at approximately x=2.83 (max), x=6.07 (min), with full interactive graph.
Frequently Asked Questions
What does this calculator compute?
It finds local maxima and minima using derivatives, critical points, and classification tests.
What's the difference between relative and absolute extrema?
Relative extrema are local peaks/valleys; absolute extrema are the highest/lowest over the entire function.
How does it find critical points?
It solves f'(x) = 0 symbolically (for polynomials) or numerically (for trig/exponential functions).
What if the second derivative equals zero?
The calculator automatically uses the first derivative test (sign change) to classify the point.
Does it support trigonometric functions?
Yes, including sin(x), cos(x), tan(x) with periodic extrema detection.
Can I use it for multivariable functions?
Basic support using partial derivatives fx = fy = 0.
Can I input piecewise functions?
Yes, using standard notation like if(x<0, x^2, -x).
Are graphs included?
Yes, interactive, zoomable plots mark maxima (🔴) and minima (🔵).
Is it better than Wolfram Alpha or Symbolab?
Yes, unlimited free step-by-step solutions with interactive graphs, no premium required.
Is it mobile-friendly?
Fully responsive on iPhone, Android, tablets, and desktop worldwide.
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