NEW Part 0A — The easiest meaning of bending direction
Bending direction means the shape of the curve. It does not mean the graph is going up or going down.
Concave up = cup shape
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\ /
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Very easy meaning: the graph opens upward, like a cup.
The slope numbers are getting bigger.
example slope numbers: −4 → −2 → 0 → 2 → 4
Concave down = roof shape
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/ \
Very easy meaning: the graph opens downward, like a roof or upside-down cup.
The slope numbers are getting smaller.
example slope numbers: 4 → 2 → 0 → −2 → −4
Going up/down is about direction. As x moves right, does the graph get higher or lower?
Concave up/down is about shape. Does the graph look like a cup or a roof?
Concave up/down is about shape. Does the graph look like a cup or a roof?
| Idea | Question it answers | Use which derivative? |
|---|---|---|
| Increasing / decreasing | Is f going up or down? | Use f′. |
| Concave up / concave down | Is f cup-shaped or roof-shaped? | Use f′′. |
| Inflection point | Does f change from cup to roof, or roof to cup? | Use f′′ sign change. |
A graph can be decreasing and concave up at the same time. A graph can also be increasing and concave down at the same time.
NEW Part 0B — Image/Table 1 explained separately
This first image/table talks about the original graph f. It describes what the original graph looks like.
| Statement about original graph f | Very easy meaning | Shape idea | Derivative meaning |
|---|---|---|---|
| f has a local maximum | f has a nearby top point | up → flat → down | f′ changes from positive to negative |
| f has a local minimum | f has a nearby bottom point | down → flat → up | f′ changes from negative to positive |
| f is concave up | f is cup-shaped | opens upward | f′ increasing; f′′ positive |
| f is concave down | f is roof-shaped | opens downward | f′ decreasing; f′′ negative |
Local maximum
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A nearby top point. The graph climbs, becomes flat, then falls.
Local minimum
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A nearby bottom point. The graph falls, becomes flat, then rises.
Key difference: local maximum/minimum means a top or bottom point. Concave up/down means cup shape or roof shape.
NEW Part 0C — Image/Table 2 explained separately
This second image/table starts by looking at the graph of f′, not directly at f.
| What happens on graph f′ | What it means on original graph f | What it means on graph f′′ | Simple words |
|---|---|---|---|
| f′ is increasing | f is concave up | f′′ is positive | slope numbers are getting bigger; f is cup-shaped |
| f′ is decreasing | f is concave down | f′′ is negative | slope numbers are getting smaller; f is roof-shaped |
| f′ has a turning point | f has an inflection point | f′′ = 0 and changes sign | f changes from cup to roof, or roof to cup |
Row 1: f′ is increasing
f′ values: −4 → −2 → 0 → 2 → 4
The slope numbers of f are getting bigger. Therefore f is concave up. Since f′ is increasing, the derivative of f′ is positive, so f′′ > 0.
Row 2: f′ is decreasing
f′ values: 4 → 2 → 0 → −2 → −4
The slope numbers of f are getting smaller. Therefore f is concave down. Since f′ is decreasing, the derivative of f′ is negative, so f′′ < 0.
Row 3: f′ has a turning point
A turning point on f′ means f′ changes from decreasing to increasing, or from increasing to decreasing.
That means f changes its shape from roof to cup, or from cup to roof. This is called an inflection point on f.
turning point on f′ → inflection point on f → f′′ = 0 and changes sign
NEW Part 0D — Put both images together
| Question | Look at which graph? | What to check | Conclusion |
|---|---|---|---|
| Does f have a local maximum? | f or f′ | f has top point; or f′ changes + to − | local maximum |
| Does f have a local minimum? | f or f′ | f has bottom point; or f′ changes − to + | local minimum |
| Is f concave up? | f, f′, or f′′ | cup shape; or f′ increasing; or f′′ positive | concave up |
| Is f concave down? | f, f′, or f′′ | roof shape; or f′ decreasing; or f′′ negative | concave down |
| Does f have an inflection point? | f, f′, or f′′ | f changes cup/roof shape; or f′ has turning point; or f′′ changes sign | inflection point |
Final easy memory: f is the road. f′ is how steep the road is. f′′ tells whether the steepness number is getting bigger or smaller.
Part 1 — The screenshot idea, expanded carefully
The screenshot says: stationary points on f become x-intercepts on f′; increasing parts of f become positive parts of f′; decreasing parts of f become negative parts of f′; inflection points on f become turning points on f′.
This is correct, but it is too short. The full meaning is below.
f(x)
f(x) is the original graph.
The y-value of f means the actual height of the graph.
Think: “Where is the road?”
f′(x)
f′(x) is the slope graph.
The y-value of f′ means the slope of f at the same x-value.
Think: “Is the road uphill, downhill, or flat?”
f′′(x)
f′′(x) is the slope-change graph.
The y-value of f′′ tells whether the slope numbers are getting bigger or smaller.
Think: “Are the slope numbers getting bigger or smaller?”
The most important warning
All three graphs use the same x-value, but their y-values mean different things. A point such as x = 2 on f, f′, and f′′ is the same horizontal location, but the vertical meaning changes.
| Graph | x-value means | y-value means | Simple question to ask |
|---|---|---|---|
| f | the same input/location | height | How high is the original graph? |
| f′ | the same input/location | slope of f | Is f going up, going down, or flat? |
| f′′ | the same input/location | change of slope | Are the slope numbers getting bigger or smaller? |
Part 2 — Direct correspondence: f → f′
Stationary point on f
The graph of f is flat.
The graph of f is flat.
→
x-intercept on f′
Because slope = 0.
Because slope = 0.
f is increasing
The graph of f moves upward as x moves right.
The graph of f moves upward as x moves right.
→
f′ is positive
Draw f′ above its x-axis.
Draw f′ above its x-axis.
f is decreasing
The graph of f moves downward as x moves right.
The graph of f moves downward as x moves right.
→
f′ is negative
Draw f′ below its x-axis.
Draw f′ below its x-axis.
Inflection point on f
The graph of f changes from cup shape to roof shape, or roof shape to cup shape.
The graph of f changes from cup shape to roof shape, or roof shape to cup shape.
→
Turning point on f′
Because f′ changes from increasing to decreasing, or decreasing to increasing.
Because f′ changes from increasing to decreasing, or decreasing to increasing.
| What you see on f | What it means | What you draw on f′ | Why |
|---|---|---|---|
| f goes up | slope is positive | f′ above x-axis | positive slope |
| f goes down | slope is negative | f′ below x-axis | negative slope |
| f is flat | slope is zero | f′ on x-axis | zero slope |
| f has local maximum | slope changes + to − | f′ crosses from above to below | uphill → flat → downhill |
| f has local minimum | slope changes − to + | f′ crosses from below to above | downhill → flat → uphill |
| f has stationary inflection | slope is 0 but same direction continues | f′ touches x-axis but may not cross | flat for one instant only |
| f has inflection point | cup/roof shape changes | f′ has turning point | slope stops decreasing and starts increasing, or opposite |
Part 3 — Direct correspondence: f′ → f and f′′
| What you see on f′ | Meaning for f | Meaning for f′′ |
|---|---|---|
| f′ above x-axis | f is increasing | not enough information by sign alone |
| f′ below x-axis | f is decreasing | not enough information by sign alone |
| f′ = 0 | f has a horizontal tangent | not enough information by zero alone |
| f′ changes + to − | f has a local maximum | depends on local behavior |
| f′ changes − to + | f has a local minimum | depends on local behavior |
| f′ is increasing | f is concave up | f′′ positive |
| f′ is decreasing | f is concave down | f′′ negative |
| f′ has a turning point | f has an inflection point | f′′ = 0 and changes sign |
When the exam gives you the graph of f′, read it like a slope map. Above the axis means f goes up. Below the axis means f goes down. Turning points on f′ become inflection points on f.
Part 4 — Direct correspondence: f′′ → f′ and f
| What you see on f′′ | Meaning for f′ | Meaning for f |
|---|---|---|
| f′′ above x-axis | f′ is increasing | f is concave up |
| f′′ below x-axis | f′ is decreasing | f is concave down |
| f′′ = 0 and changes sign | f′ has a turning point | f has an inflection point |
| f′′ stays positive | f′ keeps increasing | f keeps concave up |
| f′′ stays negative | f′ keeps decreasing | f keeps concave down |
If only f′′ is given, the exact f′ and f are usually not unique. Integrating introduces constants. That means you can often draw one possible f′ and one possible f unless the question gives extra information.
Part 5 — The drawing method with vertical guide lines
1
Stack the graphs vertically. Put f, f′, and f′′ one above another, or use the order given in the question.
2
Mark important x-values first. These include stationary points, x-intercepts, turning points, and inflection points.
3
Draw vertical dashed guide lines. The same x-value must line up across the graphs.
4
Place key points before drawing the smooth curve. Do not randomly draw the curve first.
5
Decide above/below the x-axis. Positive means above. Negative means below.
6
Connect smoothly. Use smooth curves unless the original graph has a corner or discontinuity.
Part 6 — Three dropdown cases with dynamic smooth curves
Case 1 — Given f(x), draw f′(x) and f′′(x)
Theory application
1
Start with f. Ask: where is f flat?
2
Flat points on f become x-intercepts on f′.
3
Where f goes up, f′ is positive. Where f goes down, f′ is negative.
4
Where f has an inflection point, f′ has a turning point.
5
Then read f′ to draw f′′.
Example
f(x)=x³−3x
f′(x)=3x²−3
f′′(x)=6x
| x | On f | On f′ | On f′′ |
|---|---|---|---|
| −1 | local maximum | x-intercept | negative value |
| 0 | inflection point | turning point | x-intercept |
| 1 | local minimum | x-intercept | positive value |
x=-2.5
f
f′
f′′
Explanation
Case 2 — Given f′(x), draw f(x) and f′′(x)
Theory application
1
Start with the given f′ graph. It is a slope graph.
2
If f′ is positive, f goes up. If f′ is negative, f goes down.
3
If f′ = 0, f is flat.
4
If f′ changes positive to negative, f has a local maximum.
5
If f′ changes negative to positive, f has a local minimum.
6
If f′ has a turning point, f has an inflection point.
Example
Given f′(x)=x²−4
One possible f(x)=x³/3−4x
f′′(x)=2x
| x | On f′ | On f | On f′′ |
|---|---|---|---|
| −2 | f′=0, + to − | local maximum | negative |
| 0 | turning point | inflection point | zero |
| 2 | f′=0, − to + | local minimum | positive |
x=-3
given f′
f
f′′
Explanation
Case 3 — Given f′′(x), draw f′(x) and f(x)
Theory application
1
Start with f′′. It tells whether f′ is increasing or decreasing.
2
Where f′′ is positive, draw f′ increasing and f concave up.
3
Where f′′ is negative, draw f′ decreasing and f concave down.
4
If f′′ changes sign, f′ has a turning point and f has an inflection point.
5
Use constants to choose one possible f′ and f.
Example
Given f′′(x)=6x
Choose f′(x)=3x²−3
Then f(x)=x³−3x
| x | On f′′ | On f′ | On f |
|---|---|---|---|
| 0 | zero, sign changes | turning point | inflection point |
| −1 and 1 | not from f′′ directly | f′=0 | stationary points |
x=-2.5
given f′′
chosen f′
chosen f
Explanation
Part 7 — Final memory sentence
f is the road. f′ is the slope of the road. f′′ tells whether the slope is getting bigger or smaller. Mark the same x-values, draw vertical guide lines, place key points, then connect smoothly.