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)#