AP Syllabus focus:
‘The graph of f′ links f and f″: zeros of f′ correspond to horizontal tangents of f, while slopes of f′ indicate the sign of f″ and hence concavity of f.’
The derivative links a function’s behavior to the shape of its graph, and examining f′ provides essential insight into both how f changes and how its curvature, explained by f″, behaves.
Understanding How f′ Connects f and f″
The first derivative f′ plays a central role in interpreting and predicting the behavior of its parent function f. Because f′(x) measures the instantaneous rate of change of f, its graph allows us to identify where f increases, decreases, reaches horizontal tangents, and changes concavity. This subsubtopic emphasizes two crucial relationships:
Zeros of f′ correspond to horizontal tangent lines of f, which are potential locations of local maxima, local minima, or points of inflection.
The slope of the graph of f′ tells us the sign of f″, connecting the shape of f′ directly to the concavity of f.
These connections allow us to move fluidly between the graphs of f, f′, and f″, interpreting consistent behavior across all three.
Zeros of f′ and Their Meaning for f
Horizontal Tangent Lines and Stationary Points
When the graph of f′ crosses or touches the x-axis, it satisfies f′(x)=0.
At these x-values, the tangent line to f is horizontal. These points are significant because they are:
Candidates for local extrema, where f changes direction.
Possible points of inflection, depending on concavity.
Explicit Relationship Between f′ and Tangent Behavior
Horizontal tangent line — a line touching a curve at a point where the slope equals zero.
Because f′ encodes slope information, its x-intercepts are the only locations where f could “flatten out.” Students should understand that while every local extremum requires f′(c)=0, not every zero of f′ is an extremum. The behavior of f′ around that zero will clarify which outcomes occur.
Using the Slope of f′ to Understand Concavity in f
Linking f′ to f″
The second derivative f″ represents the rate of change of f′, meaning:
When f′ is increasing, f″(x) > 0, and f is concave up.
When f′ is decreasing, f″(x) < 0, and f is concave down.
Concavity Determined From f′
Observing the slope of f′ provides immediate concavity insight:
If the graph of f′ has a positive slope on an interval, the graph of f bends upward.
If the graph of f′ has a negative slope, the graph of f bends downward.
This reinforces the idea that concavity is not just about the “shape” of f but is fundamentally tied to how the slopes of the tangent lines change over an interval.
The diagram illustrates a curve transitioning from concave up to concave down, with a labeled inflection point. Tangent segments show how slopes become less negative or less positive, connecting concavity to the changing behavior of f′(x)f'(x)f′(x). The informal mnemonics in the figure add extra detail not required by the syllabus. Source.
Key Indicators of Concavity from f′
f′ increasing → f″ positive → f concave up
f′ decreasing → f″ negative → f concave down
Essential Definitions and Equations
Concave Up: A function is concave up on an interval when the slopes of its tangent lines are increasing and f″(x) is positive on that interval.
A short explanatory sentence must follow definitions to maintain clarity. Concavity helps describe how the graph bends and helps classify extrema.
Concave Down: A function is concave down on an interval when the slopes of its tangent lines are decreasing and f″(x) is negative on that interval.
Because these definitions involve changes in slope, they reinforce why observing the slope of f′ is vital when transitioning between the first and second derivatives.
= Second derivative indicating concavity (no unit)
= First derivative indicating slope (units of f per unit x)
Understanding this relationship algebraically helps connect graphical and symbolic reasoning.
Interpreting All Three Graphs Together
Building a Consistent Picture
When analyzing function behavior, consistency among f, f′, and f″ is essential.
This graph pairs a cubic function with its quadratic derivative, illustrating that the extrema of the cubic occur where the derivative crosses the x-axis. Regions where f′(x)f'(x)f′(x) is positive or negative correspond directly to intervals where the original function is increasing or decreasing. The explicit formulas shown are extra details beyond the syllabus but align with its core ideas. Source.
The graph of f′ acts as a bridge between the visible features of f and the information encoded by f″:
Where f′ crosses the x-axis, f has a horizontal tangent.
Where f′ has positive slope, f is concave up.
Where f′ has negative slope, f is concave down.
Where f′ has a local extremum, f″(x)=0, giving possible points of inflection for f.
Practical Interpretation Strategy
Students can approach analysis in a consistent order:
Identify where f′ = 0, marking horizontal tangents for f.
Determine the sign of f′, revealing where f increases or decreases.
Examine the slope of f′, which determines concavity of f.
Connect all observations to explain the behavior of the original function.
Summary of Connections (Conceptually, not as a concluding paragraph)
Zeros of f′ → horizontal tangents of f
Sign of f′ → increasing/decreasing behavior of f
Slope of f′ → sign of f″ → concavity of f
These insights highlight why examining f′ is indispensable for understanding both the local behavior and overall shape of the function f.
FAQ
A horizontal tangent occurs where f' is zero, but the behaviour of f' around that point indicates its nature.
• If f' changes from positive to negative, f has a local maximum.
• If f' changes from negative to positive, f has a local minimum.
• If f' does not change sign, the point is neither.
This classification relies on interpreting the pattern of f' rather than the value of f itself.
Concavity depends on how the slope of f changes, and f' records that slope directly.
When the slope of f' is positive, it means the slopes of f are increasing, producing concave-up behaviour. A negative slope of f' means the slopes of f are decreasing, creating concave-down behaviour.
This makes f' a more intuitive source for concavity information than directly computing the second derivative.
Yes. Inflection points concern changes in concavity, not horizontal tangents.
A point of inflection can occur where f' has a local maximum or minimum because that is where f'' changes sign. The key requirement is that f' changes from increasing to decreasing, or vice versa.
No zero of f' is required for an inflection point to exist.
Three aspects of f' are especially useful:
• Where f' is positive or negative, indicating whether f is rising or falling.
• Where f' is zero, indicating potential turning points.
• Where f' is increasing or decreasing, indicating concavity of f.
Combining these gives a structured outline of the curve of f before worrying about fine detail.
A sharp corner occurs when f' does not exist, often shown by f' having a discontinuity or tending to different finite values from each side.
A horizontal tangent occurs when f' equals zero and the graph of f' passes smoothly through the x-axis.
Checking continuity and behaviour around that x-value helps distinguish the two cases.
Practice Questions
Question 1 (1-3 marks)
The graph of a differentiable function f is shown. At x = 3, the graph of f has a horizontal tangent.
(a) State what this tells you about the value of f'(3).
(b) Explain what feature you would expect to see at x = 3 on the graph of f'.
Question 1
(a) 1 mark:
• Correct statement that f'(3) = 0.
(b) 2 marks:
• 1 mark for stating that the graph of f' will meet or touch the x-axis at x = 3.
• 1 mark for explaining that this occurs because zeros of f' correspond to horizontal tangents of f.
Question 2 (4-6 marks)
The graph of f' for a differentiable function f is shown and is continuous on the interval 0 < x < 10.
• f' crosses the x-axis at x = 2 and x = 7.
• f' is increasing on 0 < x < 4.
• f' is decreasing on 4 < x < 10.
• f' has a local maximum at x = 4.
(a) Determine the x-values at which f has horizontal tangents and justify your answer.
(2 marks)
(b) Using the behaviour of f', state the intervals on which f is concave up and concave down.
(2 marks)
(c) Determine the x-value of any point of inflection of f and explain your reasoning.
(2 marks)
Question 2
(a) 2 marks:
• 1 mark for identifying x = 2 and x = 7 as the x-values where f has horizontal tangents.
• 1 mark for justification: these are the points where f' equals zero (crosses the x-axis).
(b) 2 marks:
• 1 mark for stating that f is concave up on 0 < x < 4 because f' is increasing there (meaning f'' is positive).
• 1 mark for stating that f is concave down on 4 < x < 10 because f' is decreasing there (meaning f'' is negative).
(c) 2 marks:
• 1 mark for identifying x = 4 as the point of inflection.
• 1 mark for justification: f' changes from increasing to decreasing at x = 4, so f'' changes sign there, indicating a change in concavity of f.
