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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 \[f'(x)=\frac{-(1+x^2)}{(x^2-1)^2} \quad \text{and}\quad f''(x) = \frac{2x(x^2+3)}{(x^2-1)^3}.\]
Given a function y = f (x), the graph is concave up (convex) in the intervals where the second derivative of the function is positive. The graph is concave down (concave) in the intervals where...
- 3 min
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- Brian McLogan
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. How to find concavity from the graph of f(x) The table below shows various graphs of f(x) and tangent lines at points x 1, x 2, and x 3. Since f'(x) is the slope of the line tangent to f(x) at point x, the concavity of f ...
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.
Of particular interest are points at which the concavity changes from up to down or down to up; such points are called inflection points. If the concavity changes from up to down at $x=a$, $f''$ changes from positive to the left of $a$ to negative to the right of $a$, and usually $f''(a)=0$.
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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.
