What are the global and local extrema of $f(x)=x^3-3x+6$ ?

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Answer 1 · 2015-12-03T14:40:37

That function has no global extrema. It has local maximum of $8$ (at ( $x=-1$ ) and local minimum of $4$ (at $x=1$ )

$lim_(xrarroo)f(x) = oo$ , so there is no global maximum.

$lim_(xrarr-oo)f(x) = -oo$ , so there is no global minimum.

$f'(x) = 3x^2-3$ which is never undefined and is $0$ at $x=-1$ and at $x=1$ . The domain of $f$ is $RR$ .

Therefore, the only critical numbers are $-1$ and $1$ .

The sign of $f'$ changes from + to - as we pass $x=-1$ , so $f(-1) = 8$ is a local maximum.

The sign of $f'$ changes from - to + as we pass $x=1$ , so $f(1) = 4$ is a local minimum.

What are the global and local extrema of f(x)=x^3-3x+6f(x)=x^3-3x+6 ?