Question

Show all the extrema and inflection points

y= (12x)/(x^2+4)

Answer 1

find first derivative and set equal to zero to find any critical points:\[f'(x)=\frac{12x^2+48-24x^2}{(x^2+4)^2}=0\]\[x^2=4\]So, critical points are at \[x=\pm2\]To see if these are local max/min try second derivative test:\[f''(x)=\frac{-24x(x^2+4)^2-(48-12x^2)2(x^2+4)(2x)}{(x^2+4)^4}\]Now try critical points to see concavity of this function at these locations:\[f''(-2)=0.75\]So x=-2 is a local minimum. And,\[f''(2)=-0.75\]So x=2 is a local max. Inflection points occur when f''(x)=0 which I'll leave for you to try.

Answer 2

Thank you!

Answer 3

np