Question

Find all points of local maxima and local minima of the function f given by $f\left(x\right)=x^3-3x+3$

Answer 1

$f\left(x\right)=x^3-3x+3$

Find all points where $f^{\prime }\left(x\right)=0$

$f^{\prime }\left(x\right)=3x^2-3=0$

$x^2-1=0$

$x=\pm 1$

Now check the sign of $f^{\prime \prime }\left(x\right)$ on these points.

$f^{\prime \prime }\left(x\right)=6x$

Now,

$f^{\prime \prime }\left(1\right)=6$ which is greater than zero.

So there is local minima at $x=1$

$f^{\prime \prime }(-1)=-6$ which is less than zero.

So there is local maxima at $x=-1$