Mastering Curver Sketching to Sketching the Curve of y=(2-x^2)/(1+x^4)

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in trying to sketch the curve of y=(2-x^2)/(1+x^4)



y'=(-2x-8x^3+2x^5)/(1+x^4)^2

y''=[ (1+x^4)^2(-2-24x^2+10x^4)-(-2x-8x^3+2x^5)8x^3(1+x^4) ] / (1+x^4)^4

=[ -2(1+x^4)(1+12x^2-5x^4)+16x^4(1+4x^2-x^4) ] /(1+x^4)^3

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to get started try plotting 2-x^2, (1+x^4) and 1/(1+x^4), this should give you a feel for the general shape of the function

noticing that the denominator is always > 0 and both the numerator & denominator are symmetric should also help
 
i need to calculate the exact points of inflections
 
hmmm... how about considering a substitution u = x^2 then

this should hopefully lead to a simpler calculation of the zeroes of the derivatives w.r.t. x

then if ' denotes dervative w.r.t. x, using the chain rule

y' = \frac{dy}{dx} = \frac{d}{dx} y(u(x)) = \frac{d y(u)}{du} \frac{du(x)}{dx} = \frac{dy}{du} u'

and so on for the next one, where you'll need the product rule as well
 
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There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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