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Curvature of the graph over an interval
- The concavity of the graph of a function refers to the curvature of the graph over an interval; this curvature is described as being concave up or concave down. Generally, a concave up curve has a shape resembling "∪" and a concave down curve has a shape resembling "∩" as shown in the figure below.
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Taking the second derivative actually tells us if the slope continually increases or decreases. When the second derivative is positive, the function is concave upward. When the second derivative is negative, the function is concave downward.
- Second Derivative
Math explained in easy language, plus puzzles, games,...
- Second Derivative
Convex curves curve downwards and concave curves curve upwards. That doesn’t sound particularly mathematical, though… When f''(x) \textcolor{purple}{> 0}, we have a portion of the graph where the gradient is increasing, so the graph is convex at this section.
The concavity of the graph of a function refers to the curvature of the graph over an interval; this curvature is described as being concave up or concave down. Generally, a concave up curve has a shape resembling "∪" and a concave down curve has a shape resembling "∩" as shown in the figure below. Concave up.
Dec 21, 2020 · The graph of a function \(f\) is concave down when \(f'\) is decreasing. That means as one looks at a concave down graph from left to right, the slopes of the tangent lines will be decreasing. Consider Figure \(\PageIndex{2}\), where a concave down graph is shown along with some tangent lines.
In mathematics, a concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to that convex combination of those domain elements. Equivalently, a concave function is any function for which the hypograph is convex.
Sal introduces the concept of concavity, what it means for a graph to be "concave up" or "concave down," and how this relates to the second derivative of a function.
Apr 18, 2023 · A concave function is a function where a line segment between any two points on the graph of the function lies entirely below the graph. Concave functions are always continuous and differentiable over their domain, have a decreasing slope, and a negative second derivative over their entire domain.
