In calculus, the first derivative tells us the slope of a curve, but it doesn't describe its shape. To understand how a graph "bends," we must look at the second derivative. The concepts of concavity and convexity allow us to describe whether a function is curving upwards like a bowl or downwards like a cap. This is essential for sketching accurate graphs and identifying the nature of stationary points in A-Level Mathematics.
Theory
The curvature of a function f(x) is determined by the rate at which its gradient changes. This rate of change is represented by the second derivative, denoted as:
or
Convex Functions (Concave Up)
A function is convex at a point or over an interval if it bends upwards. In a convex curve, any tangent line drawn to the function will lie below the curve.
- Condition: f''(x) > 0
- Visual: Think of a "cup" or a "smile."
Concave Functions (Concave Down)
A function is concave at a point or over an interval if it bends downwards. In a concave curve, any tangent line drawn to the function will lie above the curve.
- Condition: f''(x) < 0
- Visual: Think of a "cap" or a "frown."
Points of Inflection
A point of inflection occurs when a curve changes from being concave to convex, or vice versa. At this precise point, the second derivative is zero, and the concavity changes sign.
- Condition: f''(x) = 0 (and f''(x) changes sign across the point)
Steps to Find Intervals of Concavity
- Find the first derivative f'(x).
- Find the second derivative f''(x).
- Set f''(x) = 0 to find potential points of inflection.
- Test values in the intervals created by these points to check the sign of f''(x).
Worked Example
Task: Determine the intervals where the function below is concave and convex:
Step 1 - Find the first derivative:
Step 2 - Find the second derivative:
Step 3 - Solve for f''(x) = 0:
Step 4 - Test intervals: We check the sign of f''(x) for x < 2 and x > 2:
| Interval | Test Value (x) | f''(x) = 6x - 12 | Sign | Shape |
|---|---|---|---|---|
| x <2 | 0 | 6(0) - 12 = -12 | Negative | Concave |
| x > 2 | 3 | 6(3) - 12 = 6 | Positive | Convex |
Conclusion: The function is concave on the interval (-∞, 2) and convex on the interval (2, ∞).
Practice Questions & Solutions
Determine if the following quadratic function is concave or convex for all values of x:
Find the first derivative:
Find the second derivative:
Since the second derivative is a positive constant:6 > 0
The function is convex for all values of x.
Find the x-coordinate of the point of inflection for the function:
Find the first derivative:
Find the second derivative:
Set the second derivative to zero:
Over which interval is the following function concave?
Find the first derivative:
Find the second derivative:
Since the second derivative is always negative: -2 <0
The function is concave for all values of x.
Determine the concavity of the function at the point x = 1:
Find the first derivative:
Find the second derivative:
Substitute x = 1 into the second derivative:
Since the value is negative, the function is concave at x = 1.
Find the interval where the function is convex:
Find the second derivative:
For the function to be convex, the second derivative must be greater than zero:
6x - 6 > 0
6x > 6
x > 1
Summarise with AI:
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I get it somehow, there are others who are unfamiliar and I have a hard time with it but it’s okay that I get it.
Thanks for sharing your thoughts! It’s completely normal to find some parts unfamiliar or more difficult than others. The important thing is that you’re making progress and understanding more as you practise. Keep going! 😊
At least I understood some of it, but there are others that I’m confused about and have a hard time understanding, but I get it.
Thanks for sharing! It’s great that you’ve understood some of it already. Don’t worry if a few parts are still confusing — keep practising and take your time with the trickier sections. You’ll get there! 😊
The lesson was a little difficult for me, especially the word problems, but the examples helped me understand it better.
Thanks for your feedback! Word problems can definitely be tricky, so we’re glad the examples helped make the lesson clearer. Keep practising — it gets easier with time! 😊
Thankyouuu po