Search results
- The curve changes from concave to convex at x=1, so there is a point of inflexion at x = 1.
mmerevise.co.uk/a-level-maths-revision/convex-and-concave-curves/
People also ask
How do you prove a concave function?
When is a function f(x) convex (concave up)?
How do you know if a function is concave or convex?
What is a concave function?
What if a function f is concave over a convex set?
Which function is concave if f1 and f2 are convex?
In mathematics, a concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to that convex combination of those domain elements. Equivalently, a concave function is any function for which the hypograph is convex.
A function f(x) is concave (concave down) when the second derivative is negative (that is, f’’(x) < 0). Here are some examples of concave functions and their graphs. Example 1: Concave Function f(x) = -x 2
Proof. If fis concave then for any x;x 2C, x6= x, and any 2(0;1), f( x+ (1 )x) f(x) + (1 )f(x), or, dividing by and rearranging, f(x) f(x) f(x + (x x)) f(x) : Taking the limit of the right-hand side as #0 and rearranging yields inequality (7). Conversely, consider any a;b2C, take any 2(0;1), and let x = a+(1 )b. Note that a x = (1 )(b a) and b ...
- 227KB
- 12
<λf(x1) + (1 −λ)f(x2) Remark 3. A function is concave (convex) if the graph of the function is always above (below) any chord (line segment between two points in the graph). Remark 4. f concave ⇔−f convex. Example 5. Let S = [0,∞) and consider f(x) = √ x and g(x) = −f(x) = − √ x f is a concave function and g is a convex function.
Oct 24, 2024 · A function f(x) is said to be concave on an interval [a,b] if, for any points x_1 and x_2 in [a,b], the function -f(x) is convex on that interval (Gradshteyn and Ryzhik 2000).
Say we have a graph of the function f(x) = x(x^2 + 1). Find the parts of the graph where the function is convex or concave, and find the point(s) of inflexion. [3 marks] f(x) = x(x^2 + 1) = x^3 + x gives. f''(x) = 6x. f''(x) = 0, when x = 0. f''(x) \textcolor{red}{< 0} when x<0. Here we have a concave section. f''(x) \textcolor{purple}{> 0 ...
How to find concavity from the first derivative. The first derivative of a function, f' (x), is the rate of change of the function f (x). It can provide information about the function, such as whether it is increasing, decreasing, or not changing.
