Concavity and the Second Derivative
“The sign of the second derivative = the direction the curve bends.”
The formula
f''(x) > 0 ⇒ ∪ , f''(x) < 0 ⇒ ∩How to read it: if the second derivative is positive the curve is concave up (∪), if negative it's concave down (∩)
- f''(x)
- — the derivative taken twice (the second derivative)
- ∪
- — concave up (cup-shaped)
- ∩
- — concave down (cap-shaped)
The hook
Differentiate one more time and you can see which way a curve is 'bending'.
In plain words
The second derivative f'' is the rate of change of the slope. Its sign tells you whether the curve bends like a cup (∪) or a cap (∩).
The intuition
If the slope itself is increasing (f''>0), the curve bends upward into a cup shape (∪). If the slope is decreasing (f''<0), it's a cap (∩). f'' is 'the slope of the slope'.
How it's built
Differentiate once for the slope (f'), again for how that slope changes (f''). Positive sign means concave up, negative means concave down, and where the sign flips is an inflection point.
Example
For f(x)=x³, f''(x)=6x. At x=1, f''=6>0 so it's concave up; at x=−1, f''=−6<0 so concave down. The sign flips at x=0, which is the inflection point.
Common mistake
People often mislabel 'concave up (∪)' as 'concave'. Also, f''=0 doesn't guarantee an inflection point — the sign has to actually change.
Where it's used
Deciding whether a critical point is a max or a min (the second-derivative test), the exact shape of a graph, diminishing marginal utility in economics, and more.
Where it came from
The idea of curvature (how sharply something bends) goes back to Newton and Leibniz; the second derivative became the simple tool for reading the sign of that curvature.
Prerequisites
Quick check
What is the second derivative f''(x) of f(x)=x²?
- 2x
- 2✓
- 0
- x²
Practice
What is f''(2) for f(x)=x³?
Answer: 12
- f'(x)=3x²
- f''(x)=6x
- Plug in x=2 → 12
Takeaway: Differentiate twice to get the second derivative.
What is f''(1) for f(x)=x⁴?
Answer: 12
- f'(x)=4x³
- f''(x)=12x²
- Plug in x=1 → 12
Takeaway: f''>0, so near x=1 the curve is concave up.
Find the x-coordinate of the inflection point of f(x)=x³−3x².
Answer: undefined
- f'(x)=3x²−6x
- f''(x)=6x−6
- Set to 0: 6x−6=0 → x=1 (the sign flips around it, so it's an inflection point)
Takeaway: An inflection point is where f''=0 and its sign changes.