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Is #f(x)=-x^5-2x^2-6x+3# concave or convex at #x=-4#?

Answer 1

Ezra Adams

The function is convex

Calculate the first and second derivatives

#f(x)=-x^5-2x^2-6x+3#

#f'(x)=-5x^4-4x-6#

#f''(x)=-20x^3-4#

Therefore, when #x=-4#

#f''(-4)=-20*(-4)^3-4=1280-4=1276#

As #f''(x)>0#, the function is convex at #x=-4#

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Answer 2

Christopher Allers

To determine the concavity of the function ( f(x) = -x^5 - 2x^2 - 6x + 3 ) at ( x = -4 ), we need to find the second derivative of the function and evaluate it at ( x = -4 ).

The first derivative of ( f(x) ) is ( f'(x) = -5x^4 - 4x - 6 ).

The second derivative of ( f(x) ) is ( f''(x) = -20x^3 - 4 ).

Now, plug in ( x = -4 ) into the second derivative:

[ f''(-4) = -20(-4)^3 - 4 ] [ = -20(-64) - 4 ] [ = - (-1280) - 4 ] [ = 1280 - 4 ] [ = 1276 ]

Since the second derivative ( f''(-4) = 1276 ) is positive, the function is concave upwards at ( x = -4 ).

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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.