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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
Dec 21, 2020 · Let \(f(x)=x/(x^2-1)\). Find the inflection points of \(f\) and the intervals on which it is concave up/down. Solution. We need to find \(f'\) and \(f''\). Using the Quotient Rule and simplifying, we find \[f'(x)=\frac{-(1+x^2)}{(x^2-1)^2} \quad \text{and}\quad f''(x) = \frac{2x(x^2+3)}{(x^2-1)^3}.\]
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.
How to find the concavity of a function. 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 ...
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 the second derivative of a function is positive, then the function is concave upwards. If the second derivative of a function is negative, then the function is concave downwards.
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Feb 1, 2024 · Find the second derivative ($f”(x)$) of the function. Use a number line to test the sign of the second derivative at various intervals. A positive $f”(x)$ indicates the function is concave up; the graph lies above any drawn tangent lines, and the slope of these lines increases with successive increments.
