Introduction
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:
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
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