Curve sketching
Shape from signs
If you knew only where f' and f'' are positive and negative, could you sketch f? Which features of the graph would you know exactly?
The three graphs below share one x-axis. Drag the ochre cursor across them and compare what each graph is doing at the same x. Shading marks where f' and f'' are positive (ochre) or negative (indigo).
y = f(x)
y = f′(x): positive means f rises
y = f″(x): positive means f bends upward
What each derivative tells you
- First derivative
f'>0 on an interval means f is increasing there, and f'<0 means decreasing. A critical number is where f'=0 or f' does not exist.
- First derivative test
If f' changes from positive to negative at a critical number, f has a local maximum there. From negative to positive gives a local minimum. No change in sign gives neither.
- Second derivative
f''>0 means f is concave up, bending like a cup. f''<0 means concave down. A point where the concavity changes is an inflection point.
- Second derivative test
If f'(c)=0 and f''(c)>0, there is a local minimum at c. If f''(c)<0, a local maximum. If f''(c)=0, the test is inconclusive.
Turn off "Show f″" and try predicting where the inflection points of f are, using only f'. Where on the graph of f' do they appear?
Solve it step by step
Find the derivatives, read their signs, and describe the shape. The score counts steps you answer correctly on the first try.
Common mistakes
- Assuming f′ = 0 means a maximum or minimum
x^3 has f'(0)=0 but keeps rising. Check that the sign of f' actually changes.
- Assuming f″ = 0 means an inflection point
x^4 has f''(0)=0 but is concave up on both sides. The concavity must change.
- Confusing concave down with decreasing
A graph can rise while bending downward. The first and second derivatives answer different questions.