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When is a twice-differentiable function f concave down?
What if a function is not twice differentiable at C?
Why is a function f concave up if f ′ is decreasing?
Can the second derivative test be used on a function that is twice differentiable?
Which function is concave down over the interval?
What if f is continuous and F changes concavity at a?
Dec 21, 2020 · Let \(f\) be twice differentiable on an interval \(I\). The graph of \(f\) is concave up if \(f''>0\) on \(I\), and is concave down if \(f''<0\) on \(I\).
Therefore, a twice-differentiable function [latex]f[/latex] is concave down when [latex]f^{\prime \prime}(x)<0[/latex]. Applying this logic is known as the concavity test. Test for Concavity
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. If f ″ (x) is positive on an interval, the graph of y = f(x) is concave up on that interval.
Aug 19, 2023 · By definition, a function \(f\) is concave up if \(f'\) is increasing. From Corollary \(3\), we know that if \(f'\) is a differentiable function, then \(f'\) is increasing if its derivative \(f''(x)>0\). Therefore, a function \(f\) that is twice differentiable is concave up when \(f''(x)>0\).
Learning Objectives. By the end of this section, the student should be able to: Describe how the second derivative of a function relates to its concavity and how to apply the second derivative test. Describe the relationship between inflection points and concavity and how to find the inflection points of a function. Second Derivative and Concavity.
The graph of a function \(f\) is concave down when \(\fp \)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 3.4.3 , where a concave down graph is shown along with some tangent lines.
From Corollary 3, 3, we know that if f ′ f ′ is a differentiable function, then f ′ f ′ is increasing if its derivative f ″ (x) > 0. f ″ (x) > 0. Therefore, a function f f that is twice differentiable is concave up when f ″ (x) > 0. f ″ (x) > 0. Similarly, a function f f is concave down if f ′ f ′ is decreasing.
