kaliprasad said:I was reading some where that $\sqrt{x}$ is concave function what does it mean.
That is, of course true, and often the easiest way to use "convex function", but is not the definition of "convex function". A set is "convex" if and only if, given any two points, A and B, in that set the line segment between A and B is also in the set. A function, f, is said to be "convex" ("convex upward" is typically implied by "convex" alone) if and only if the set of all points above the graph of y= f(x) is a convex set. The function is "convex downward" if the set of all points below the graph of y= f(x) is a convex set.MarkFL said:I had always thought of a concave function, or concave down, over some interval, as having a negative second derivative on that interval. If we have:
\(\displaystyle f(x)=\sqrt{x}\)
then we find:
\(\displaystyle f''(x)=-\frac{1}{4x^{\frac{3}{2}}}\)
Hence, we see that on the interval $(0,\infty)$ we have $f''<0$.
A concave function is a type of mathematical function that has a curved shape, resembling a bowl. It is a function that decreases at a decreasing rate, meaning the rate of change decreases as the input value increases.
A concave function is the opposite of a convex function. While a concave function decreases at a decreasing rate, a convex function increases at an increasing rate. The shape of a convex function is like a hill, while a concave function is like a bowl.
Concave functions are important in mathematics because they have many useful applications, such as optimization problems in economics and engineering. They also have properties that make them easier to work with, such as being differentiable at every point.
One way to identify a concave function is by looking at its graph. If the graph of a function is curved downward, it is likely a concave function. Another way is by calculating the second derivative of the function. If the second derivative is negative, the function is concave.
Yes, a concave function can have multiple points of inflection. These are points where the concavity changes from concave up to concave down, or vice versa. The number of points of inflection depends on the complexity of the function and the range of the input values.