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- If the second derivative of a function is positive over an interval, the function is concave up on that interval, meaning it curves upward. Conversely, if the second derivative is negative over an interval, the function is concave down, indicating it curves downward.
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Dec 21, 2020 · When the graph is concave up, the critical point represents a local minimum; when the graph is concave down, the critical point represents a local maximum. We have been learning how the first and second derivatives of a function relate information about the graph of that function.
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
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.
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.
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.
Definition. Analyzing concavity involves determining the direction in which a function curves, specifically whether it is concave up (curving upwards) or concave down (curving downwards).
Definition 3.4.1 Concave Up and Concave Down. Let \(f\) be differentiable on an interval \(I\text{.}\) The graph of \(f\) is concave up on \(I\) if \(\fp\) is increasing. The graph of \(f\) is concave down on \(I\) if \(\fp\) is decreasing. If \(\fp\) is constant then the graph of \(f\) is said to have no concavity. Loose Language
