How do you determine where the given function #f(x) = (x+3)^(2/3)  6# is concave up and where it is concave down?
In order to investigate concavity, we'll look at the sign of the second derivative.
graph{(x+3)^(2/3)  6 [18.74, 13.3, 15.11, 0.92]}
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To determine where the given function ( f(x) = (x+3)^{\frac{2}{3}}  6 ) is concave up and where it is concave down, we need to analyze the second derivative of the function.

Find the first derivative of ( f(x) ). [ f'(x) = \frac{2}{3}(x+3)^{\frac{1}{3}} ]

Find the second derivative of ( f(x) ). [ f''(x) = \frac{2}{9}(x+3)^{\frac{4}{3}} ]

Determine where the second derivative is positive (concave up) and where it is negative (concave down) by examining the sign of ( f''(x) ).
 ( f''(x) > 0 ): Concave up
 ( f''(x) < 0 ): Concave down
Since the second derivative is always negative for all values of ( x ), the function ( f(x) ) is concave down for all ( x ).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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