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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.
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.
Convex curves, or convex functions are curves that curve upwards. Graphically, if a curve is above the line segment connecting any two points on it, the curve is said to be convex. Conversely, concave functions curve downwards.
Graphically, a concave function opens downward, and water poured onto the curve would roll off. A function f is convex if f’’ is positive (f’’ > 0). A convex function opens upward, and water poured onto the curve would fill it. Of course, there is some interchangeable terminology at work here.
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.
Convex and Concave functions and inflection points. Convex and concave are words that we use to describe the shape or curvature of a curve. Recall from classifying stationary points (see Stationary Points page) that we can find the second derivative of a function by differentiating twice.
Nov 16, 2022 · The second derivative will allow us to determine where the graph of a function is concave up and concave down. The second derivative will also allow us to identify any inflection points (i.e. where concavity changes) that a function may have.
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