Is #f(x)=x^5-4x^4+16x-4# concave or convex at #x=-2#?

Answer 1

The answer is #f(x)# is concave at #x=-2#

Let's calculate the first and second derivatives

#f(x)=x^5-4x^4+16x-4#
#f'(x)=5x^4-16x^3+16#
#f''(x)=20x^3-48x^2=4x^2(5x-12)#
Now, we calculate, #f''(-2)#
#f''(-2)=20*(-2)^3-48*(-2)^2#
#=-160-192=-352#
As, #f''(-2)<0#, we conclude that #f(-2)# is concave #nn#

Normally, we calculate the concavity over an interval for example

#]-oo, 0[#
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Answer 2

To determine if the function ( f(x) = x^5 - 4x^4 + 16x - 4 ) is concave or convex at ( x = -2 ), we need to analyze the second derivative of the function at that point.

  1. Find the first derivative of ( f(x) ): [ f'(x) = 5x^4 - 16x^3 + 16 ]

  2. Find the second derivative of ( f(x) ): [ f''(x) = 20x^3 - 48x^2 ]

  3. Evaluate the second derivative at ( x = -2 ): [ f''(-2) = 20(-2)^3 - 48(-2)^2 = -160 ]

Since the second derivative is negative at ( x = -2 ), the function is concave downward at that point.

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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.

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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