Concavity is a concept in calculus that describes the "curvature" or the "bending direction" of a function’s graph. To determine whether a function is concave up or concave down at a certain interval, we use the second derivative of the function.

1. Concavity and Curvature

• Concave Up: A function is concave up on an interval if its graph lies above its tangent lines and "bends" upwards like a cup. This happens when the slope of the tangent lines (represented by the first derivative) is increasing as you move along the graph.

• Concave Down: A function is concave down on an interval if its graph lies below its tangent lines and "bends" downwards like a frown. This occurs when the slope of the tangent lines is decreasing along the graph.

2. The Role of the Second Derivative

The second derivative of a function, $f''(x)$, tells us the rate of change of the first derivative $f'(x)$, which represents the slope of the tangent line.

• If $f''(x) > 0$ on an interval, then $f(x)$ is concave up on that interval.
• If $f''(x) < 0$ on an interval, then $f(x)$ is concave down on that interval.

3. Inflection Points

An inflection point occurs where the concavity of a function changes, meaning the graph switches from being concave up to concave down or vice versa. For an inflection point to exist at $x = c$, the second derivative should change sign at $c$ (from positive to negative or vice versa).

Example

Suppose we have a function $f(x) = x^3 - 3x$. Here’s a breakdown:

1. First Derivative: $f'(x) = 3x^2 - 3$
2. Second Derivative: $f''(x) = 6x$

• For $x > 0$, $f''(x) = 6x > 0$, so $f(x)$ is concave up.
• For $x < 0$, $f''(x) = 6x < 0$, so $f(x)$ is concave down.

The function changes concavity at $x = 0$, which is therefore an inflection point.

Would you like further clarification or have a specific example in mind?

Related Questions

1. What does the second derivative test tell us about local maxima and minima?
2. How is the first derivative related to concavity?
3. Can concavity be applied to multidimensional functions?
4. Why are inflection points significant in graph analysis?
5. How does the second derivative affect the shape of a graph in real-world applications?

Tip

To quickly determine concavity in an exam, check the sign of the second derivative at key points within intervals. Positive means concave up, and negative means concave down.