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- We say that a graph is concave up if the line between two points is above the graph, or alternatively if the first derivative is increasing. (In finance, such a curve is said to be convex.) Similarly, we say that a graph is concave down if the line between two points is below the graph, or alternatively if the first derivative is decreasing.
mathstat.slu.edu/~may/ExcelCalculus/sec-4-5-SecondDerivativeConcavity.htmlThe Second Derivative and Concavity - Saint Louis University
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Example 1: Characterising Graphs. Say we have a graph of the function f(x) = x(x^2 + 1). Find the parts of the graph where the function is convex or concave, and find the point(s) of inflexion. [3 marks] f(x) = x(x^2 + 1) = x^3 + x gives. f''(x) = 6x. f''(x) = 0, when x = 0. f''(x) \textcolor{red}{< 0} when x<0. Here we have a concave section.
Dec 21, 2020 · A function is concave down if its graph lies below its tangent lines. If knowing where a graph is concave up/down is important, it makes sense that the places where the graph changes from one to the other is also important.
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- Definition 2
- Fact
- Second Derivative Test
To show that the graphs above do in fact have concavity claimed above here is the graph again (blown up a little to make things clearer). So, as you can see, in the two upper graphs all of the tangent lines sketched in are all below the graph of the function and these are concave up. In the lower two graphs all the tangent lines are above the graph...
Now that we have all the concavity definitions out of the way we need to bring the second derivative into the mix. We did after all start off this section saying we were going to be using the second derivative to get information about the graph. The following fact relates the second derivative of a function to its concavity. The proof of this fact ...
So, what this fact tells us is that the inflection points will be all the points where the second derivative changes sign. We saw in the previous chapter that a function may change signs if it is either zero or does not exist. Note that we were working with the first derivative in the previous section but the fact that a function possibly changing ...
The third part of the second derivative test is important to notice. If the second derivative is zero then the critical point can be anything. Below are the graphs of three functions all of which have a critical point at x=0x=0, the second derivative of all of the functions is zero at x=0x=0and yet all three possibilities are exhibited. The first i...
Dec 21, 2020 · If we are trying to understand the shape of the graph of a function, knowing where it is concave up and concave down helps us to get a more accurate picture. Of particular interest are points at which the concavity changes from up to down or down to up; such points are called inflection points.
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.
The definition of the concavity of a graph is introduced along with inflection points. Examples, with detailed solutions, are used to clarify the concept of concavity. Example 1: Concavity Up. Let us consider the graph below. Note that the slope of the tangent line (first derivative) increases. The graph in the figure below is called concave up .
If \( f''(x) \) is positive on an interval, the graph of \( y=f(x) \) is concave up on that interval. We can say that \(f\) is increasing (or decreasing) at an increasing rate. If \( f''(x) \) is negative on an interval, the graph of \( y=f(x) \) is concave down on that interval.
