Question

Find all the points of local maxima and local minima of the function $f.$
$f\left(x\right)=2x^3-6x^2+6x+5$

Answer 1

$f\left(x\right)=2x^3-6x^2+6x+5$

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

$\to f^{\prime }\left(x\right)=\frac{d(2x^3-6x^2+6x+5}{dx}$

$f^{\prime }\left(x\right)=6x^2-12x+6+0$

$f^{\prime }\left(x\right)=6(x^2-2x+1)$

$f^{\prime }\left(x\right)=6(x^2+1^2-2(x\left)\right(1\left)\right)=6(x-1{)}^2$

put $f^{\prime }\left(x\right)=0$

$\to 6(x-{)}^2=0\to x=1$

At $x=1$ if $x<|$,say $0.9,$ Sign of $f^{\prime }\left(x\right)>0$ maxima or minima since sign of $f^{\prime }\left(x\right)$ does not change ,it is point of inflexion$

At $x=1$ if $x<|$,say $1.1,$ Sign of $f^{\prime }\left(x\right)>0$ maxima or minima since sign of $f^{\prime }\left(x\right)$does not change ,it is point of inflexion

Hence $x=1$is point of inflexion

$x=1$,is neither maxima nor point of minima.