For #f(x)=x^3 + 4x^2 - 5x # ,
#lim_(xrarr-oo)f(x)=-oo# so #f# has no global minimum.
#lim_(xrarroo)f(x)=oo# so #f# has no global maximum.
#f'(x)=3x^2+8x-5# is never undefined and is #0# at
#x=(-4+-sqrt31)/3#
For numbers far from #0# (both positive and negative), #f'(x)# is positive.
For numbers in #((-4-sqrt31)/3,(-4+sqrt31)/3)#, 3f'(x)# is negative.
The sign of #f'(x)# changes from + to - as we move past #x=(-4-sqrt31)/3#, so #f((-4-sqrt31)/3)# is a local maximum.
The sign of #f'(x)# changes from - to + as we move past #x=(-4+sqrt31)/3#, so #f((-4+sqrt31)/3)# is a local minimum.
Finish by doing the arithmetic to get the answer:
#f# has a local maximum of #f((-4-sqrt31)/3) = (308+62sqrt31)/27# and a local minimum of #f((-4+sqrt31)/3) = (308-62sqrt31)/27#
