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blaah
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Homework Statement
f:R->R, c'
prove that f is concave iff f(x*)+(x-x*)f'(x*)>=f(x)
Homework Equations
assume the function is only once differentiable
The Attempt at a Solution
i have no idea how to approach this question...
Are x* and x any two values of x? Are there any restrictions on the values of x?blaah said:Homework Statement
f:R->R, c'
prove that f is concave iff f(x*)+(x-x*)f'(x*)>=f(x)
Homework Equations
assume the function is only once differentiable
The Attempt at a Solution
i have no idea how to approach this question...
A function is concave if it has a graph that curves downward, resembling a bowl. This means that as the input increases, the rate of change of the function decreases. In other words, the function is "curving inward" and has a decreasing slope.
To prove that a function is concave, you can use the second derivative test. This involves taking the derivative of the function twice and examining the sign of the second derivative at different points. If the second derivative is negative at all points, then the function is concave.
No, a function cannot be both convex and concave at the same time. This is because a convex function has a graph that curves upward, while a concave function has a graph that curves downward. These two types of functions are considered opposites and cannot coexist.
Not necessarily. A function can be concave on a certain interval and discontinuous at certain points within that interval. However, if a function is concave on a closed interval, it must be continuous at all points within that interval.
Knowing that a function is concave can be useful in various applications, such as optimizing functions in economics or engineering. It can also help in understanding the behavior of a function and predicting its future values. Additionally, concave functions have many properties that make them easier to work with in mathematical calculations.