# On what interval is #f(x)=6x^3+54x-9# concave up and down?

A function is concave up when the second derivative is positive, it is concave down when it is negative, and there could be an inflection point when it is zero.

so:

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To determine the intervals where ( f(x) = 6x^3 + 54x - 9 ) is concave up and concave down, we need to find the second derivative of the function and analyze its sign.

First, find the first derivative of ( f(x) ) with respect to ( x ): [ f'(x) = 18x^2 + 54 ]

Next, find the second derivative of ( f(x) ): [ f''(x) = 36x ]

To determine concavity, analyze the sign of ( f''(x) ):

- If ( f''(x) > 0 ), the function is concave up.
- If ( f''(x) < 0 ), the function is concave down.

Since ( f''(x) = 36x ), it's positive when ( x > 0 ) (concave up) and negative when ( x < 0 ) (concave down).

Therefore, ( f(x) = 6x^3 + 54x - 9 ) is concave up for ( x > 0 ) and concave down for ( x < 0 ).

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