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- The graph of f is concave up on I if f ′ is increasing. The graph of f is concave down on I if f ′ is decreasing.
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Is the graph of (F) concave down on (I)?
Is f(x) f (x) concave down on an interval?
Is (F) concave up?
What is a concave graph?
How do you know if a function is concave up or down?
Can a second derivative determine concavity without a graph?
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.
Nov 16, 2022 · If \(f''\left( x \right) < 0\) for all \(x\) in some interval \(I\) then \(f\left( x \right)\) is concave down on \(I\). So, what this fact tells us is that the inflection points will be all the points where the second derivative changes sign.
If f' (x) is decreasing over an interval, then the graph of f (x) is concave down over the interval. Given a graph of f (x) or f' (x), as well as the facts above, it is relatively simple to determine the concavity of a function.
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.
Oct 22, 2024 · For the function \(f(x)=x^3−6x^2+9x+30,\) determine all intervals where \(f\) is concave up and all intervals where \(f\) is concave down. List all inflection points for \(f\). Use a graphing utility to confirm your results.
If [latex]f''(x)[/latex] is positive on an interval, the graph of [latex]y=f(x)[/latex] is concave up on that interval. We can say that [latex]f[/latex] is increasing (or decreasing) at an increasing rate. If [latex]f''(x)[/latex] is negative on an interval, the graph of [latex]y=f(x)[/latex] is concave down on that interval.
Review your knowledge of concavity of functions and how we use differential calculus to analyze it.
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