What are the global and local extrema of $f(x)=2x^7-2x^5$ ?

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Answer 1 · 2015-12-28T16:13:15

We rewrite f as

$f(x)=2x^7*(1-1/x^2)$

but $lim_(x->oo) f(x)=oo$ hence there is no global extrema.

For the local extrema we find the points where $(df)/dx=0$

$f'(x)=0=>14x^6-10x^4=0=>2*x^4*(7*x^2-5)=0=>x_1=sqrt(5/7) and x_2=-sqrt(5/7)$

Hence we have that

local maximum at $x=-sqrt(5/7)$ is $f(-sqrt(5/7))=100/343*sqrt(5/7)$

and

local minimum at $x=sqrt(5/7)$ is $f(sqrt(5/7))=-100/343*sqrt(5/7)$

What are the global and local extrema of f(x)=2x^7-2x^5 f(x)=2x^7-2x^5 ?