What are the absolute extrema of # f(x)= x^(2)+2/x # on the interval [1,4]?

1 Answer

We need to find the critical values of #f(x)# in the interval #[1,4]#.

Hence we calculate the roots of the first derivative so we have

#(df)/dx=0=>2x-2/x^2=0=>2x^2(x-2)=0=>x=2#

So #f(2)=5#

Also we find the values of #f# at the endpoints hence

#f(1)=1+2=3#

#f(4)=16+2/4=16.5#

The largest function value is at #x=4# hence #f(4)=16.5# is the absolute maximum for #f# in #[1,4]#

The smallest function value is at #x=1# hence #f(1)=3# is the absolute minimum for #f# in #[1,4]#

The graph of #f# in #[1,4]# is

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