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Dec 21, 2020 · The number line in Figure \(\PageIndex{5}\) illustrates the process of determining concavity; Figure \(\PageIndex{6}\) shows a graph of \(f\) and \(f''\), confirming our results. Notice how \(f\) is concave down precisely when \(f''(x)<0\) and concave up when \(f''(x)>0\).
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.
Review your knowledge of concavity of functions and how we use differential calculus to analyze it.
- You have two functions f and g, where f''>0 and g''>0. If you take the second derivative of f+g, you get f''+g'', which is positive. So their sum i...
- In order for 𝑓(𝑥) to be concave up, in some interval, 𝑓 ''(𝑥) has to be greater than or equal to 0 (i.e. non-negative) for all 𝑥 in that inter...
- In general, no. Points where f"(x)= 0 are specifically called inflection points. But in the example provided, -2 is also a critical point (Observe...
- Product rule: d/dx[f(x)*g(x)] = f'(x)g(x) + f(x)g'(x) f(x) = (x-1)³ and g(x) = x
There are a number of ways to determine the concavity of a function. If given a graph of f (x) or f' (x), determining concavity is relatively simple. Otherwise, the most reliable way to determine concavity is to use the second derivative of the function; the steps for doing so as well as an example are located at the bottom of the page.
2. Concavity. THE EFFICIENCY OF A WORKER. In the preceding section, you saw how to use the sign of the derivative f (x) to deter-mine where f(x) is increasing and decreasing and where its graph has relative extrema. In this section, you will see that the second derivative f (x) also provides useful infor-mation about the graph of f(x).
Apr 24, 2022 · 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.
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.
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