Math · Calculus I · Concept
Curve sketching with derivatives
The first derivative tells you where a function increases or decreases, and its sign changes at critical points locate local maxima and minima. The second derivative tells you where the graph is concave up or down, and its sign changes locate inflection points. Together they give the graph’s shape.
Increasing and decreasing
Where f′(x) > 0 the graph rises; where f′(x) < 0 it falls. The sign of f′ can change only at critical points, where f′ is zero or undefined, so testing one number in each interval between them gives the sign on the whole interval.
The first derivative test
At a critical point c, if f′ changes from positive to negative, f has a local maximum; if it changes from negative to positive, a local minimum. If f′ keeps the same sign on both sides, there is no extremum, even though f′(c) = 0.
| f′ left of c | f′ right of c | At c |
|---|---|---|
| + | − | Local maximum |
| − | + | Local minimum |
| + | + | No extremum |
| − | − | No extremum |
Concavity and the second derivative
Where f″(x) > 0 the graph is concave up, bending like a cup, and its slopes increase. Where f″(x) < 0 it is concave down. A point where the concavity changes is an inflection point.
The second derivative test
At a critical point with f′(c) = 0, f″(c) < 0 means a local maximum and f″(c) > 0 a local minimum. If f″(c) = 0, the test gives no answer; go back to the first derivative test.
Putting it together
A full sketch uses the domain, the intercepts, the critical points, the signs of f′ and f″, and any asymptotes. Plot the key points first, then join them with the right rise, fall and bend on each interval.
Common mistakes
- Calling every point where f′ = 0 an extremum: x⁴ − 4x³ has f′(0) = 0 but no maximum or minimum at 0.
- Calling every zero of f″ an inflection point: the concavity must actually change.
- Using the sign of f″ to decide where f increases: that is the job of f′.
- Testing only one side of a critical point.
Key terms
- Critical point
- A point in the function’s domain where the derivative is zero or doesn’t exist. Local maxima and minima can happen only at critical points or endpoints, so these are where to look.
- First derivative test
- A way to classify a critical point by the sign of f′ on either side: + to − gives a local maximum, − to + a local minimum, and no change of sign means no extremum.
- Second derivative test
- At a critical point c with f′(c) = 0, f″(c) < 0 gives a local maximum and f″(c) > 0 a local minimum. If f″(c) = 0 the test gives no answer, and the first derivative test decides.
- Concavity
- The direction a graph bends. Where f″ > 0 the graph is concave up and its slopes increase; where f″ < 0 it is concave down and its slopes decrease.
- Inflection point
- A point on a continuous graph where the concavity changes. There f″ is zero or undefined, but a zero of f″ is not enough on its own: the sign of f″ must change.
- Local maximum
- A function value at least as large as all the nearby values. It doesn’t have to be the largest value overall.
- Local minimum
- A function value at most as large as all the nearby values. At an endpoint, compare only with the values on the side that exists.
Work through an example
Find where f(x) = x³ − 3x² − 9x + 5 increases and decreases, its local extrema, its concavity and its inflection point.
Sketch a cubic with its derivatives →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.
Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
Open the Derivative Tracer Graph f and its key points in Graph Check the derivatives in Math Open worked example on a board Derivative tests in Math ReferenceYour existing work stays on this device. Examples open as editable copies.