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

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Answer 1 · 2016-03-18T17:27:19

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

enter image source here

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