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- Concavity in a curve refers to its curvature, or the way it bends. If a curve is concave up, it opens upward like a cup, while if it's concave down, it opens downward like a frown. Mathematically, a curve is concave up if its second derivative is positive, and concave down if its second derivative is negative.
www.geeksforgeeks.org/concavity-and-points-of-inflection/
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Dec 21, 2020 · Definition Concave Up and Concave Down. Let \(f\) be differentiable on an interval \(I\). 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. If \(f'\) is constant then the graph of \(f\) is said to have no concavity.
Given any x 1 or x 2 on an interval such that x 1 < x 2, if f (x 1) > f (x 2), then f (x) is decreasing over the interval. In the graph of f' (x) below, the graph is decreasing from (-∞, 1) and increasing from (1, ∞), so f (x) is concave down from (-∞, 1) and concave up from (1, ∞).
Sal introduces the concept of concavity, what it means for a graph to be "concave up" or "concave down," and how this relates to the second derivative of a function.
- From that point on, the slope goes from being negative to becoming zero. Hence, it stops decreasing (in other words, it increases till it becomes z...
- Increasing slope can mean one of two things: _more_ positive or _less_ negative. Whichever situation you have, increasing slope always implies conc...
- Short answer: no. Since the function f is not defined by some formula, only by the graph sal draw, you cant say wether or not these are parabolas....
- A point where both f''(x) = 0 and f''(x) changes sign (i.e. f(x) changes concavity) is called a point of inflection of f(x). Visually, the graph of...
- Observe that he was looking at the graph of f'(x). This graph is the derivative of the first graph. So, in the first graph, at that point, the deri...
- When f'(x) is positive, f(x) increases When f'(x) is negative, f(x) decreases When f'(x) is zero, it indicates a *possible* local max or min (use t...
Theorem 3. Let C R be an open interval. 1. f: C!R is concave i for any a;b;c2C, with a<b<c, f(b) f(a) b a f(c) f(b) c b; and, f(b) f(a) b a f(c) f(a) c a: For strict concavity, the inequalities are strict. 2. f: C!R is convex i for any a;b;c2C, with a<b<c, f(b) f(a) b a f(c) f(b) c b; and, f(b) f(a) b a f(c) f(a) c a:
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Dec 21, 2020 · Example \(\PageIndex{1}\) Describe the concavity of \( f(x)=x^3-x\). Solution. The first dervative is \( f'(x)=3x^2-1\) and the second is \(f''(x)=6x\). Since \(f''(0)=0\), there is potentially an inflection point at zero.
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.
Example 5.4.1 Describe the concavity of $\ds f(x)=x^3-x$. First, we compute $\ds f'(x)=3x^2-1$ and $f''(x)=6x$. Since $f''(0)=0$, there is potentially an inflection point at zero.
