Is #f(x)=(x3)^3x+15# concave or convex at #x=3#?
Neither.
You have to find the second derivative to answer this question.
Now it's easy enough to expand to find the second derivative using the power rule.
Now find the second derivative.
We have:
Hopefully this helps!
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To determine whether ( f(x) = (x  3)^3  x + 15 ) is concave or convex at ( x = 3 ), we need to examine the second derivative of the function at that point.

Find the first derivative of ( f(x) ): [ f'(x) = 3(x  3)^2  1 ]

Find the second derivative of ( f(x) ): [ f''(x) = 6(x  3) ]

Evaluate the second derivative at ( x = 3 ): [ f''(3) = 6(3  3) = 0 ]
Since the second derivative at ( x = 3 ) is equal to zero, the function neither concave nor convex at that point. Instead, it exhibits an inflection point.
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To determine whether the function ( f(x) = (x  3)^3  x + 15 ) is concave or convex at ( x = 3 ), we can analyze the sign of the second derivative of the function at that point.

Find the first derivative of ( f(x) ): [ f'(x) = 3(x  3)^2  1 ]

Find the second derivative of ( f(x) ): [ f''(x) = 6(x  3) ]

Evaluate the second derivative at ( x = 3 ): [ f''(3) = 6(3  3) = 0 ]
Since the second derivative ( f''(3) ) is equal to zero, we cannot determine the concavity or convexity of the function at ( x = 3 ) using this method. Further analysis or another method may be needed to determine the concavity or convexity at that point.
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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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 What is the second derivative of #f(x)= 1/x^3x^3#?
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