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What is a concave graph?
How do you know if a graph is convex or concave?
What does a concave up curve look like?
How do you know if a function is concave?
Is the graph of (F) concave down on (I)?
Can a second derivative determine concavity without a graph?
When f''(x) \textcolor{red}{< 0}, we have a portion of the graph where the gradient is decreasing, so the graph is concave at this section. An easy way to test for both is to connect two points on the curve with a straight line. If the line is above the curve, the graph is convex. If the line is below the curve, the graph is concave.
Dec 21, 2020 · The graph of \(f\) is concave up if \(f''>0\) on \(I\), and is concave down if \(f''<0\) on \(I\). Figure \(\PageIndex{3}\): Demonstrating the 4 ways that concavity interacts with increasing/decreasing, along with the relationships with the first and second derivatives.
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.
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.
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.
- From that point on, the slope goes from being negative to becoming zero. Hence, it stops decreasing (in other words, it increases till it becomes z...
- Increasing slope can mean one of two things: _more_ positive or _less_ negative. Whichever situation you have, increasing slope always implies conc...
- Short answer: no. Since the function f is not defined by some formula, only by the graph sal draw, you cant say wether or not these are parabolas....
- A point where both f''(x) = 0 and f''(x) changes sign (i.e. f(x) changes concavity) is called a point of inflection of f(x). Visually, the graph of...
- Observe that he was looking at the graph of f'(x). This graph is the derivative of the first graph. So, in the first graph, at that point, the deri...
- When f'(x) is positive, f(x) increases When f'(x) is negative, f(x) decreases When f'(x) is zero, it indicates a *possible* local max or min (use t...
If f ′ (x) is positive on an interval, the graph of y = f(x) is increasing on that interval. If f ′ (x) is negative on an interval, the graph of y = f(x) is decreasing on that interval. The second derivative tells us if a function is concave up or concave down.
State the first derivative test for critical points. Use concavity and inflection points to explain how the sign of the second derivative affects the shape of a function’s graph. Explain the concavity test for a function over an open interval.
