Critical Point (calculus)
A critical point of a function is any point in its domain where the derivative is either zero or undefined, making it a candidate location for local maxima, local minima, or saddle points. Critical points are the primary tool for optimisation in calculus, used to find the maximum or minimum values of a function on an interval. They are fundamental to applications in economics (profit maximisation), engineering (structural design), and physics (equilibrium states).
Key Formula
f'(c) = 0 or f'(c) is undefined, where c is a critical point
LaTeX: f'(c) = 0 \quad \text{or} \quad f'(c) \text{ does not exist}
| Symbol | Meaning | Unit |
|---|---|---|
| c | the critical point value of x | dimensionless |
| f'(c) | first derivative of f evaluated at c | dimensionless |
Worked Example
Problem
Find the critical points of f(x) = x³ − 6x² + 9x + 1 and classify them.
Solution
Step 1: Find f'(x): f'(x) = 3x² − 12x + 9. Step 2: Set f'(x) = 0: 3x² − 12x + 9 = 0. Step 3: Divide by 3: x² − 4x + 3 = 0. Step 4: Factor: (x − 1)(x − 3) = 0, so x = 1 and x = 3. Step 5: Find f''(x) = 6x − 12. Step 6: At x = 1: f''(1) = 6 − 12 = −6 < 0 → local maximum. Step 7: At x = 3: f''(3) = 18 − 12 = 6 > 0 → local minimum. Step 8: f(1) = 1 − 6 + 9 + 1 = 5; f(3) = 27 − 54 + 27 + 1 = 1.
Answer
Local maximum at (1, 5); local minimum at (3, 1)
Classifying Critical Points Using the Second Derivative Test
| Condition | Classification | Graph Shape | Example |
|---|---|---|---|
| f'(c) = 0 and f''(c) < 0 | Local maximum | Concave down at c | Hilltop |
| f'(c) = 0 and f''(c) > 0 | Local minimum | Concave up at c | Valley bottom |
| f'(c) = 0 and f''(c) = 0 | Inconclusive (use first deriv. test) | May be inflection or extremum | x³ at x = 0 |
| f'(c) undefined | Potential cusp or corner | Sharp point in graph | |x| at x = 0 |
| f'(c) = 0 (endpoint) | Absolute max/min candidate | Endpoint of closed interval | Optimisation on [a, b] |
Interactive Tools
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Related Terms
Derivative (calculus)
The derivative of a function at a point measures the instantaneous rate of change of the function's output with respect to its input at that point, and geometrically represents the slope of the tangent line to the function's graph. Derivatives are defined as the limit of the difference quotient as the interval shrinks to zero. They are central to physics, engineering, economics, and all sciences wherever rates of change or optimisation are relevant.
Differentiation
Differentiation is the process of computing the derivative of a function, yielding a new function that expresses the rate of change of the original at every point in its domain. It involves applying systematic rules — such as the power rule, product rule, chain rule, and quotient rule — to transform a given function into its derivative. Differentiation is used extensively in physics for velocity and acceleration, in economics for marginal analysis, and in engineering for optimisation and control systems.
Related Rates
Related rates problems use implicit differentiation with respect to time to find how the rate of change of one quantity relates to the rate of change of another quantity, given that both are functions of time. A geometric or physical relationship between the quantities is differentiated with respect to time, and known rates are substituted to find the unknown rate. These problems are extensively used in physics and engineering, for instance to relate the rate at which a ladder slides down a wall to the rate at which the base moves away from the wall.
The word "critical" comes from the Greek "kritikos" meaning able to discern or judge, and the Latin "criticus." In calculus, critical points are so named because they are decisive or judgemental points where the behaviour of a function may change — coined in the context of optimisation theory in the 18th and 19th centuries.