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Concave Up

Concave Up

A graph or part of a graph which looks like a right-side up bowl or part of an right-side up bowl.

A coordinate plane with x and y axes showing a curve that opens upward like a bowl, labeled "concave up.

 

 

See also

Concave down, concave

Key Formula

f′′(x)>0  ⟹  f is concave upf''(x) > 0 \implies f \text{ is concave up}
Where:
  • f(x)f(x) = The original function
  • f′′(x)f''(x) = The second derivative of f, which measures how the slope is changing

Worked Example

Problem: Determine where f(x) = x² − 4x + 3 is concave up.
Step 1: Find the first derivative of f(x).
f′(x)=2x−4f'(x) = 2x - 4
Step 2: Find the second derivative of f(x).
f′′(x)=2f''(x) = 2
Step 3: Determine where the second derivative is positive. Since f''(x) = 2 for all values of x, the second derivative is always positive.
f′′(x)=2>0 for all xf''(x) = 2 > 0 \text{ for all } x
Answer: The function f(x) = x² − 4x + 3 is concave up on the entire real number line (−∞, ∞). This makes sense because the graph is an upward-opening parabola — a perfect bowl shape.

Another Example

Problem: Find the intervals where g(x) = x³ − 6x² + 9x + 1 is concave up.
Step 1: Compute the first derivative.
g′(x)=3x2−12x+9g'(x) = 3x^2 - 12x + 9
Step 2: Compute the second derivative.
g′′(x)=6x−12g''(x) = 6x - 12
Step 3: Set the second derivative greater than zero and solve.
6x−12>0  ⟹  x>26x - 12 > 0 \implies x > 2
Step 4: State the interval of concavity. The function is concave up when x > 2, meaning on the interval (2, ∞). At x = 2, the concavity changes — this point is called an inflection point.
g′′(x)>0 on (2,∞)g''(x) > 0 \text{ on } (2, \infty)
Answer: g(x) = x³ − 6x² + 9x + 1 is concave up on the interval (2, ∞).

Frequently Asked Questions

How do you tell if a graph is concave up or concave down?
If the graph curves upward like a bowl (or a smile), it is concave up. If it curves downward like an upside-down bowl (or a frown), it is concave down. Using calculus, check the second derivative: if f''(x) > 0, the function is concave up; if f''(x) < 0, it is concave down.
Can a function be concave up and decreasing at the same time?
Yes. Concavity and direction (increasing or decreasing) are independent properties. For example, f(x) = x² is decreasing on (−∞, 0) but concave up everywhere. The curve goes downhill but still bends upward, like the left side of a bowl.

Concave Up vs. Concave Down

A function is concave up when it bends upward (f''(x) > 0) and concave down when it bends downward (f''(x) < 0). Think of concave up as a bowl that holds water, and concave down as a hill that sheds water. The point where a function switches between concave up and concave down is called an inflection point.

Why It Matters

Concavity tells you how a function is curving, which goes beyond just knowing whether it rises or falls. In optimization problems, concave up at a critical point confirms you have found a minimum — this is the second derivative test. In real-world modeling, concavity reveals whether a rate of change (like speed, profit, or population growth) is accelerating or decelerating.

Common Mistakes

Mistake: Confusing concave up with increasing. Students assume that if a curve is concave up, it must be going upward.
Correction: Concavity describes how the curve bends, not which direction it travels. A function can be concave up while decreasing — picture the left side of a U-shaped parabola.
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. If f''(x) > 0, the function is concave up.

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