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

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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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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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.

- What are the points of inflection, if any, of #f(x)= x^3 + 9x^2 + 15x - 25 #?
- How do you use the first and second derivatives to sketch #f(x) = sqrt(4 - x^2)#?
- How do you find the first and second derivative of #(lnx)/x^2#?
- What is the second derivative of #f(x)=(e^x + e^-x) / 2 #?
- What is the second derivative of #f(x)= sqrt(5+x^6)/x#?

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