# Is #f(x)=-3x^3+4x^2+3x-4# concave or convex at #x=-1#?

It is convex.

graph{-3x^3+4x^2+3x-4 [-9.96, 10.04, -6.12, 3.88]}

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To determine the concavity or convexity of ( f(x) = -3x^3 + 4x^2 + 3x - 4 ) at ( x = -1 ), we need to examine the sign of the second derivative, ( f''(x) ), at that point.

First, find the first derivative ( f'(x) ) by differentiating ( f(x) ): [ f'(x) = -9x^2 + 8x + 3 ]

Now, find the second derivative ( f''(x) ) by differentiating ( f'(x) ): [ f''(x) = -18x + 8 ]

Evaluate ( f''(-1) ): [ f''(-1) = -18(-1) + 8 = -18 + 8 = -10 ]

Since ( f''(-1) = -10 ) is negative, the function is concave down at ( x = -1 ).

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When evaluating a one-sided 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 one-sided 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 one-sided 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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- How do you find the extremas for #f(x)=x^3 - 9x^2 + 27x#?
- How do you determine the interval(s) on which the function #y= ln(x) / x^3# is concave up and concave down?

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