For what values of x is #f(x)=4/(x^2-1# concave or convex?
Please see the explanation below
The function is
Calulate the first derivative with the quotient rule
Therefore,
Calulate the second derivative with the quotient rule
Therefore,
Build a variation chart to determine the concavities
Finally,
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To determine the concavity of ( f(x) = \frac{4}{x^2 - 1} ), we need to find its second derivative and then analyze its sign.
First, find the first derivative:
[ f'(x) = \frac{d}{dx} \left( \frac{4}{x^2 - 1} \right) = \frac{-8x}{(x^2 - 1)^2} ]
Now, find the second derivative:
[ f''(x) = \frac{d}{dx} \left( \frac{-8x}{(x^2 - 1)^2} \right) = \frac{-8(x^2 - 1)^2 - (-8x)(2(x^2 - 1)(2x))}{(x^2 - 1)^4} = \frac{-8(x^2 - 1) + 16x^2}{(x^2 - 1)^3} ] [ = \frac{-8x^2 + 8 + 16x^2}{(x^2 - 1)^3} = \frac{8 - 8x^2}{(x^2 - 1)^3} ]
To find the intervals of concavity, set ( f''(x) > 0 ) for convex and ( f''(x) < 0 ) for concave.
[ \frac{8 - 8x^2}{(x^2 - 1)^3} > 0 ]
[ 8 - 8x^2 > 0 ]
[ 8(1 - x^2) > 0 ]
[ (1 - x^2) > 0 ]
[ 1 > x^2 ]
[ -1 < x < 1 ]
Therefore, the function ( f(x) = \frac{4}{x^2 - 1} ) is concave for ( -1 < x < 1 ) and convex elsewhere.
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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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