Home > Calculus > Analyzing Concavity of a Function

Is #f(x)=-x^5+3x^4-9x^3-2x^2-6x# concave or convex at #x=-3#?

Answer 1

Emma Aldridge

*Answer could only be described as concave UPWARDS or DOWNWARDS (in your case, I'm not sure on how you view convex)

Answer: f''(-3) = 1022, hence, concave UPWARD

Method used: Second derivative test

Take the second derivative of f(x). #f''(x)=-20x^3+36x^2-54x-4#

Then evaluate f(x) at x = -3

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Answer 2

Emma Aldridge

To determine the concavity of the function ( f(x) = -x^5 + 3x^4 - 9x^3 - 2x^2 - 6x ) at ( x = -3 ), we need to evaluate the second derivative of the function at that point.

First, find the first derivative of ( f(x) ): [ f'(x) = -5x^4 + 12x^3 - 27x^2 - 4x - 6 ]

Now, find the second derivative: [ f''(x) = -20x^3 + 36x^2 - 54x - 4 ]

Evaluate ( f''(-3) ): [ f''(-3) = -20(-3)^3 + 36(-3)^2 - 54(-3) - 4 ] [ f''(-3) = -20( -27 ) + 36(9) + 162 - 4 ] [ f''(-3) = 540 + 324 + 162 - 4 ] [ f''(-3) = 1022 ]

Since ( f''(-3) ) is positive (specifically, greater than zero), the function is concave upward 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.