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Is #f(x)=-2x^3-2x^2+8x-1# concave or convex at #x=3#?

Answer 1

Cameron Alexander

Concave (sometimes called "concave down")

Concavity and convexity are determined by the sign of the second derivative of a function:

If #f''(3)<0#, then #f(x)# is concave at #x=3#.
If #f''(3)>0#, then #f(x)# is convex at #x=3#.
To find the function's second derivative, use the power rule repeatedly.

#f(x)=-2x^3-2x^2+8x-1#
#f'(x)=-6x^2-4x+8#
#f''(x)=-12x-4#

The value of the second derivative at #x=3# is

#f''(3)=-12(3)-4=-40#

Since this is #<0#, the function is concave at #x=3#:

These are the general shapes of concavity (and convexity):

We can check the graph of the original function at #x=3#:

graph{-2x^3-2x^2+8x-1 [-4,4, -150, 40]}

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Answer 2

Isaiah Allen

To determine whether ( f(x) = -2x^3 - 2x^2 + 8x - 1 ) is concave or convex at ( x = 3 ), we need to analyze the second derivative of the function.

The second derivative of ( f(x) ) is ( f''(x) = -12x - 4 ).

Evaluate ( f''(3) ): [ f''(3) = -12(3) - 4 = -36 - 4 = -40 ]

Since ( f''(3) ) is negative, the function is concave at ( x = 3 ).

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Answer from HIX Tutor

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.