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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...

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- Thread starter blaah
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- #1

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f:R->R, c'

prove that f is concave iff f(x*)+(x-x*)f'(x*)>=f(x)

assume the function is only once differentiable

i have no idea how to approach this question...

- #2

Mark44

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Are x* and x any two values of x? Are there any restrictions on the values of x?## 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...

To prove your statement you need to prove two things:

- f is concave ==> f(x*) + (x - x*) f'(x*) >= f(x)
- f(x*) + (x - x*) f'(x*) >= f(x) ==> f is concave

For the second, one approach would be a proof by contradiction. Suppose that f(x*) + (x - x*) f'(x*) >= f(x) is true and that f is not concave. If you arrive at a contradiction, it means that your original assumption was incorrect, and therefore f must be concave.

Mark

- #3

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i know that for the function to be concave all the points on the tangent need to be on or below the function....but i doesn't help....i've been staring at the problems for days now, with no result...

- #4

HallsofIvy

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Looks to me like the mean value theorem would be useful here.

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