Gradient graphs, concavity and inflection tests
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| point of inflection/pɔɪnt ɒv ɪnˈflekʃn/ | điểm uốn |
Which gradient graph tells the difference?
- Two curves can both have zero slope, yet one turns and the other keeps rising.
- This lesson studies point of inflection 拐点: A point where a continuous curve changes its direction of concavity.
The method
- The sign of f′ identifies increasing or decreasing intervals.
- Zeros of f′ are stationary candidates.
- A change + to − gives a local maximum; − to + gives a local minimum.
- The second derivative f″ measures how the gradient changes: positive means an increasing gradient and a convex (concave-up) curve; negative means a decreasing gradient and a concave (concave-down) curve.
- An inflection needs a change of concavity, not only f″=0.
For q=x⁴, q″(0)=0. Why is (0,0) NOT a point of inflection?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Inflection needs a CHANGE of concavity - a sign change in f″. The zero of the second derivative alone is inconclusive; inspect signs each side, as x³ does change sign at 0 while x⁴ does not.
A worked case
For f=x³−3x, f′=3x²−3 and f″=6x. The gradient graph is above zero for x<−1 or x>1 and below for −1<x<1. Thus (−1,2) is a local maximum and (1,−2) a local minimum. The second derivative changes from negative to positive at x=0, so (0,0) is an inflection with nonzero gradient −3. For g=x³, g′=3x² stays positive on both sides of zero: (0,0) is a stationary inflection, not a turning point. For q=x⁴, q′=4x³ changes − to + and q″=12x² is nonnegative: (0,0) is a minimum, but not an inflection even though q″(0)=0.
Gradient graphs, concavity and inflection tests
The sign of f′ identifies increasing or decreasing intervals
Connect a derivative calculation to its limit or gradient sign interpretation.
For x³−3x, find the local maximum y-coordinate.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
At x=−1 the derivative changes + to − and f(−1)=2.
Where it goes wrong
- Do not confuse the height of f with its gradient f′.
- A decreasing curve can have an increasing gradient while staying below zero in the gradient graph.
- The test f″(a)=0 is inconclusive; inspect signs on each side.
- A point of inflection need not be stationary, and a stationary point need not be an extremum.
- Endpoint extrema on a restricted interval also require checking the endpoints.
Not true: Every point at which the second derivative is zero is a point of inflection. At an inflection the concavity must change; y=x⁴ has a zero second derivative at 0 but no inflection.
For x³−3x, find f″(1).
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Differentiate twice: f″=6x, so f″(1)=6.
At an inflection the concavity must change.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. At an inflection the concavity must change; y=x⁴ has a zero second derivative at 0 but no inflection.
Applying it
- Align the x-axes of the original curve and gradient graph.
- Mark zeros of f′ and build sign intervals before naming extrema.
- Read concavity from whether the gradient graph rises or falls.
- For a twice-differentiable polynomial, use an actual sign change of f″ to establish an inflection and evaluate the original function for its coordinates.
For x⁴, find q″(0).
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
q″=12x² is zero at zero; this alone does not imply an inflection.
Match each derivative sign pattern to its classification.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The first derivative locates candidates; sign changes - of f′ for extrema, of f″ for inflections - do the classifying.
In your course
- AQA A-level Mathematics (7357), specification section G.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Point of inflection: A point where a continuous curve changes its direction of concavity.
- The sign of f′ identifies increasing or decreasing intervals.
- Watch for: do not confuse the height of f with its gradient f′.