Key takeaway: Concavity describes the direction a graph bends: concave up where the slope is increasing and the second derivative is positive, concave down where the slope is decreasing and it is negative.
Grade 12 – AP Calculus ABAP Calculus AB Unit 5Calculus
Definition of Concavity
Concavity tells whether a curve bends like a cup or like a cap. On a concave up stretch the curve lies above its tangent lines, and its slope keeps growing from left to right. On a concave down stretch the curve lies below its tangent lines, and its slope keeps shrinking.
Because the second derivative is the rate of change of the slope, its sign reports the bend. Concavity says nothing about whether the function itself rises or falls; a decreasing function can be concave up.
Formula and notation
On an interval where the second derivative exists:
\[ y^{\prime\prime}\gt 0\ \Rightarrow\ \text{concave up},\qquad y^{\prime\prime}\lt 0\ \Rightarrow\ \text{concave down}. \]
The interval boundaries come from zeros of \(y^{\prime\prime}\) and from places where it is undefined.
Key facts
- Concave up resembles a cup that holds water; concave down resembles a cap that spills it.
- Concavity is about the slope changing, not about the function increasing or decreasing.
- A graph that is concave up lies above each of its tangent lines.
- Test one number in each interval between the zeros of the second derivative.
Concavity example
Find where \(f(x)=x^4-6x^2\) is concave up and where it is concave down.
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First derivative: \(f^{\prime}(x)=4x^3-12x\). Second derivative: \(f^{\prime\prime}(x)=12x^2-12=12(x-1)(x+1)\).
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The second derivative is zero at \(x=-1\) and \(x=1\), which cut the line into three intervals.
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Test \(x=-2\): \(f^{\prime\prime}=36\), positive. Test \(x=0\): \(f^{\prime\prime}=-12\), negative. Test \(x=2\): \(36\), positive.
The graph is concave up on \((-\infty,-1)\) and \((1,\infty)\), and concave down on \((-1,1)\).
Common mistake
Watch out
A student sees that \(f^{\prime}(x)=4x^3-12x\) is positive at \(x=2\) and concludes that the graph is concave up there because it is rising.
Why it is wrong
Rising and concave up are separate ideas: the first comes from the sign of \(f^{\prime}\), the second from the sign of \(f^{\prime\prime}\). Here both happen to be positive at \(x=2\), but a curve can fall while concave up or rise while concave down. Always check the second derivative for concavity.
Related terms and lessons
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