Clear["Global`*"]

Поиск точек экстремума и перегиба функции

f[x_]:=x^3-3x+1
Solve[f'[x]==0,x]
Solve[f''[x]==0,x]

f''[-1]
f''[1]

{f[-1],f[1],f[0]}

Plot[f[x],{x,-3,3},PlotRange->{-5,5}]

График функции и ее производных первого и второго порядка

f[x_]:=x^3-2x^2-x+4;
xmax=1/3(2-Sqrt[7]);
xmin=1/3(2+Sqrt[7]);
xp=2/3;
TabView[{
"f[x]"->Show[
Plot[{f[x],f'[x],f''[x]},{x,-3,3},PlotRange->{-7,7},PlotStyle->{Line,Dashed,DotDashed},PlotLegends->"Expressions"],
ListPlot[{
{{xmax,f[xmax]},{xmin,f[xmin]},{xp,f[xp]}}
}
,PlotStyle->Red
]
],
"f'[x]"->Show[
Plot[{f[x],f'[x],f''[x]},{x,-3,3},PlotRange->{-7,7},PlotStyle->{Line,Dashed,DotDashed},PlotLegends->"Expressions"],
ListPlot[{
{{xmax,f[xmax]},{xmin,f[xmin]},{xp,f[xp]}},
{{xmax,f'[xmax]},{xmin,f'[xmin]},{xp,f'[xp]}}
}
,PlotStyle->Red,
Filling->{2->{{1},Blue}}
]
],
"f''[x]"->Show[
Plot[{f[x],f'[x],f''[x]},{x,-3,3},PlotRange->{-7,7},PlotStyle->{Line,Dashed,DotDashed},PlotLegends->"Expressions"],
ListPlot[{
{{xmax,f[xmax]},{xmin,f[xmin]},{xp,f[xp]}},
{{xmax,f'[xmax]},{xmin,f'[xmin]},{xp,f'[xp]}},
{{xmax,f''[xmax]},{xmin,f''[xmin]},{xp,f''[xp]}}
}
,PlotStyle->Red,
Filling->{2->{{1},Blue}, 3->{{1},Blue}}
]
]
}]