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May 22, 2024 · Solved Examples on Concavity. Example 1: Determine the intervals where the function f(x)=x 3 −6x 2 +9x+15 is concave up and concave down. First Derivative: f'(x)=3x 2-12x + 9. Second Derivative: f"(x)=6x-12. Find Critical Points for Concavity: Set the second derivative equal to zero to find potential points of inflection: 6x-12=0. x=12/6=2
- Concave Function
Solved Examples on Concave Function. Practice Questions on...
- Convex and Concave Functions
Solved Problems on Convex and Concave Functions. Problem 1:...
- Concave Function
- What Are Concave functions?
- Solved Examples on Concave Function
- Practice Questions on Concave Function
- Summary
A Concave function is also called a Concave downward graph. Intuitively, the Concavity of the function means the direction in which the function opens, concavity describes the state or the quality of a Concave function. For example, if the function opens upwards it is called concave up and if it opens downwards it is called concave down. The figure...
Example 1: What should be the value of “a” for the function f(x) = ax3+ 4x2+ 1 to be concave downward at x = 1. Solution: Example 2: What is the shape of the graph for the function f(x) =1x+4\frac{1}{x + 4} x+41at x = 2. Solution: Example 3: What is the shape of the graph for the function f(x) = x2+ x + 1at x = 0. Solution: Example 4: What is the ...
1. Prove that f(x) = -x² is a concave function. 2. Is the function f(x) = ln(x) concave? Justify your answer. 3. Show that the sum of two concave functions is also concave. 4. Determine if f(x) = √x is concave on its domain. 5. Prove or disprove: If f(x) is concave, then -f(x) is convex. 6. Is the function f(x) = 1/x concave on (0, ∞)? Explain your...
A concave function is a fundamental concept in mathematics, particularly in optimization and analysis. In a single variable context, a function f(x) is concave if its graph lies above or on any line segment connecting two points on the graph. More formally, for any two points x₁ and x₂ in the domain and any t ∈ [0,1], a concave function satisfies f...
Sep 22, 2024 · Solved Problems on Convex and Concave Functions. Problem 1: Show that f(x) = e x is a convex function. Solution: First, compute the second derivative of f(x): f'(x) = e x and f''(x) = e x. Since f''(x) = e x ≥ 0 for all x , f(x) is convex. Problem 2: Determine if g(x) = -x 2 is concave or convex. Solution: First, compute the second derivative:
Jul 30, 2019 · ax.scatter(hull_pts[0], hull_pts[1], color='red') ax.add_patch(PolygonPatch(hull, fill=False, color='green')) One possible solution is to take each line and interpolate it to a range of let's say 20 points and find the concave hull of all the created points.
A concave hull may be the solution for some real-world problems (e.g. finding the reasonable boundary of a city). Is there a proper definition, algorithm and practical solution for the notion of a Concave Hull?
Example. Let's see an example of concave function: Commonly, we can say that the convex functions are curved functions that are first decreasing and afterwards increasing, while the concave functions are the other way round, they are first increasing and afterwards increasing.
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1. Consider the two points (x1, f(x1)) and (x2, f(x2)). The function f between x1 and x2 is no lower than the line segment connecting these two points. For example, λ = 0.5 corresponds to the halfway point. The LHS corresponds to the midpoint of the line segment, the RHS corresponds to the function evaluated at the average of x1 and x2.
