Is #f(x)=x^5-4x^4+16x-4# concave or convex at #x=-2#?
The answer is
Let's calculate the first and second derivatives
Normally, we calculate the concavity over an interval for example
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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.
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Find the first derivative of ( f(x) ): [ f'(x) = 5x^4 - 16x^3 + 16 ]
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Find the second derivative of ( f(x) ): [ f''(x) = 20x^3 - 48x^2 ]
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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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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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