Concave Down
Concave Down
A graph or part of a graph which looks like an upside-down bowl or part of an upside-down bowl.
See also
Key Formula
f′′(x)<0
Where:
- f′′(x) = The second derivative of f(x), which measures how the slope is changing
- x = Any x-value in the interval being tested
Worked Example
Problem: Determine where the function f(x) = -x² + 4x is concave down.
Step 1: Find the first derivative of f(x).
f′(x)=−2x+4
Step 2: Find the second derivative of f(x).
f′′(x)=−2
Step 3: Check the sign of the second derivative. Since f''(x) = -2, which is negative for all values of x, the condition f''(x) < 0 holds everywhere.
−2<0for all x
Step 4: Conclude the concavity. Because the second derivative is always negative, the graph is an upside-down parabola that opens downward.
Answer: f(x) = -x² + 4x is concave down on the entire real line, (-∞, ∞).
Another Example
Problem: Find the intervals where g(x) = x³ - 6x² + 9x + 1 is concave down.
Step 1: Find the first derivative.
g′(x)=3x2−12x+9
Step 2: Find the second derivative.
g′′(x)=6x−12
Step 3: Set the second derivative less than zero and solve for x.
6x−12<0⟹x<2
Step 4: The graph is concave down wherever x < 2. At x = 2, the concavity changes — this is called an inflection point.
Answer: g(x) is concave down on the interval (-∞, 2).
Frequently Asked Questions
How do you tell if a graph is concave up or concave down?
If the graph curves like a bowl that holds water (opens upward), it is concave up. If it curves like an upside-down bowl (opens downward), it is concave down. Using calculus, check the second derivative: f''(x) > 0 means concave up, and f''(x) < 0 means concave down.
Can a function be concave down and still be increasing?
Yes. Concavity describes how the slope changes, not whether the function goes up or down. For example, f(x) = -x² + 4x is both increasing and concave down on the interval (0, 2). The function rises, but it rises at a decreasing rate.
Concave Down vs. Concave Up
A concave down curve bends downward and has a decreasing slope (f''(x) < 0), shaped like an upside-down bowl or the top of a hill. A concave up curve bends upward and has an increasing slope (f''(x) > 0), shaped like a right-side-up bowl or the bottom of a valley. The point where a curve switches between concave up and concave down is called an inflection point.
Why It Matters
Concavity helps you understand the full shape of a graph, not just whether it goes up or down. In optimization problems, the second derivative test uses concavity to determine whether a critical point is a maximum (concave down) or a minimum (concave up). In real-world contexts, concavity describes how a rate of change itself is changing — for example, whether economic growth is accelerating or slowing.
Common Mistakes
Mistake: Confusing concave down with decreasing. Students assume that if a curve is concave down, the function must be going down.
Correction: Concavity describes how the slope changes, not the direction of the function. A function can be increasing while concave down — it just means the rate of increase is slowing.
Mistake: Using the first derivative instead of the second derivative to test concavity.
Correction: The first derivative tells you whether the function is increasing or decreasing. You need the second derivative to determine concavity: f''(x) < 0 means concave down.
Related Terms
- Concave Up — Opposite concavity — curve bends upward
- Concave — General term for the shape of a curve
- Graph of an Equation or Inequality — Visual representation where concavity is observed
- Inflection Point — Where concavity changes from up to down or vice versa
- Second Derivative — The test used to determine concavity
- Second Derivative Test — Uses concavity to classify maxima and minima