Is #f(x)=1e^(3x)# concave or convex at #x=2#?
Find the second derivative at
If the second derivative is positive , it's concave up . Likewise, if it is negative , it will be concave down .
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To determine whether ( f(x) = 1  e^{3x} ) is concave or convex at ( x = 2 ), we need to analyze the second derivative of the function at that point.

Find the second derivative of the function: [ f(x) = 1  e^{3x} ] [ f'(x) = 3e^{3x} ] [ f''(x) = 9e^{3x} ]

Evaluate the second derivative at ( x = 2 ): [ f''(2) = 9e^{3(2)} = 9e^6 ]
Since the second derivative is negative at ( x = 2 ), the function is concave downwards at that point.
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To determine whether the function ( f(x) = 1  e^{3x} ) is concave or convex at ( x = 2 ), we need to analyze the second derivative of the function at that point.
First, find the second derivative of the function: [ f(x) = 1  e^{3x} ] [ f'(x) = 3e^{3x} ] [ f''(x) = 9e^{3x} ]
Now evaluate the second derivative at ( x = 2 ): [ f''(2) = 9e^{3(2)} = 9e^6 ]
Since the second derivative is negative at ( x = 2 ), ( f(x) ) is concave downward at ( x = 2 ).
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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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