Find intervals where the graph of the function f(x)= x^3-9x is concave up or concave down and all inflection points.

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The intervals where a function is concave up is where the second derivative is positive, and it is concave down when the second derivative is negative.

`f(x)=x^3-9x`

`f'(x)=3x^2-9`

`f''(x)=6x`

We see that the second derivative is positive when `x>0` and negative when `x<0` .  This also means that there is only one inflection point at (0,0).

The function has an inflection point at (0,0), is concave up for `x>0` and concave down for `x<0` .

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