For what values of x is #f(x)=4/(x^2-1# concave or convex?

Answer 1

Please see the explanation below

The function is

#f(x)=4/(x^2-1)#
The domain of #f(x)# is #x in (-oo, -1)uu(-1, 1)uu(1, +oo)#

Calulate the first derivative with the quotient rule

#(u/v)'=(u'v-uv')/(v^2)#
#u=4#, #=>#, #u'=0#
#v=x^2-1#, #=>#, #v'=2x#

Therefore,

#f'(x)=(0*(x^2-1)-4*2x)/(x^2-1)^2=-(8x)/(x^2-1)^2#
#f'(x)=0#, #=>#, #x=0#
There is a critical point at #(0, -4)#

Calulate the second derivative with the quotient rule

#u=-8x#, #=>#, #u'=-8#
#v=(x^2-1)^2#, #=>#, #v'=4x(x^2-1)#
#f''(x)=(-8(x^2-1)^2+32x^2(x^2-1))/(x^2-1)^4#
#=(-8x^2+8+32x^2)/(x^2-1)^3#
#=(24x^2+8)/(x^2-1)^3#

Therefore,

#f''(x)!=0#, #AA x in "domain"#

Build a variation chart to determine the concavities

#color(white)(aaaa)##"Interval"##color(white)(aaaa)##(-oo,-1)##color(white)(aaaa)##(-1,1)##color(white)(aaaa)##(1,+oo)#
#color(white)(aaaa)##"Sign f''(x)"##color(white)(aaaaaaa)##+##color(white)(aaaaaaaaaa)##-##color(white)(aaaaaaaa)##+#
#color(white)(aaaa)##" f(x)"##color(white)(aaaaaaaaaaa)##uu##color(white)(aaaaaaaaaa)##nn##color(white)(aaaaaaaa)##uu#

Finally,

#f(x)# is convex for #x in (-oo,-1)uu(1, +oo)#
#f(x)# is concave for #x in (-1,1)#
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Answer 2

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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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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