How to define the judgment function of extremum points

Question

I encountered a problem in solving the math problem in China's 2019 postgraduate entrance examination.

I need to judge whether $f(x)=left{begin{array}{cc}
x|x|, & x leq 0 
x ln x, & x>0
end{array}right. $ is differentiable at the point x = 0 and whether it is an extreme point.
The code in this post can be used to judge whether it is differentiable or not:
differentiableQ[f_, spec : (v_ -> v0_)] := 
  With[{jac = D[f, {v}]}, 
   Module[{f0, jac0}, {f0, jac0} = {f, jac} /. Thread[spec];
     VectorQ[Flatten@{f0, jac0}, NumericQ] && 
      Limit[(f - f0 - jac0.(v - v0))/Sqrt@Total[(v - v0)^2], spec] ===
        0] /; VectorQ[jac]];
ClearAll[differentiableQ, dLimit];
differentiableQ[f_, spec : (v_ -> v0_)] := 
  With[{jac = D[f, {v}]}, 
   Module[{f0, jac0, res}, {f0, jac0} = {f, jac} /. Thread[spec];
     If[VectorQ[Flatten@{f0, jac0}, NumericQ], 
      res = Limit[(f - f0 - jac0.(v - v0))/Sqrt@Total[(v - v0)^2], 
         spec] /. 
        HoldPattern[Limit[df_, s_]] /; ! FreeQ[df, Piecewise] :> 
         With[{L = dLimit[df, s]}, L /; FreeQ[L, dLimit]];
      res = 
       FreeQ[res, Indeterminate] && And @@ Thread[Flatten@{res} == 0],
       res = False]] /; VectorQ[jac]];
dLimit[df_, spec_] := 
  Module[{f0, jac0, pcs = {}, z, res}, 
   pcs = Replace[(*Solve[..,Reals] separates PW fn*)
     z /. Solve[z == df, z, 
       Reals], {ConditionalExpression[y_, c_] :> {y, c}, 
      y_ :> {y, True}}, 1];
   If[ListQ[pcs], 
    res = (Limit[Piecewise[{#}], spec] /. 
         HoldPattern[Limit[Piecewise[{{y_, _}}, 0], s_]] :> 
          With[{L = Limit[y, s]}, L /; FreeQ[L, Limit]] & /@ pcs)];
   res /; ListQ[pcs]];

f[x_] := Piecewise[{{x*RealAbs[x], x <= 0}, {x*Log[x], x > 0}}]
differentiableQ[f[x], {x} -> {0}]

Now I want to judge whether the point x = 0 is the extreme point of function $f(x)=left{begin{array}{cc}
x|x|, & x leq 0 
x ln x, & x>0
end{array}right. $ .
However, MMA has no built-in function to determine whether a point is an extreme point. How to define a custom function to determine the extreme point?

Bob Hanlon · Answer

Since Abs cannot be differentiated and you are dealing with real x, either change Abs[x] to Sqrt[x^2] Clear["Global`*"] f[x_] = Piecewise[{{x*Abs[x], x <= 0}, {x*Log[x], x > 0}}] /. Abs[real_] :> Sqrt[real^2]; f'[x] // FullSimplify Or, alternatively use RealAbs f2[x_] = Piecewise[{{x*RealAbs[x], x <= 0}, {x*Log[x], x > 0}}]; f2'[x] Consequently, the derivative is not defined at x == 0 Graphically, Plot[{f'[x], f[x]}, {x, -2, 2}, PlotLegends -> Placed["Expressions", {0.7, 0.3}]]

How to define the judgment function of extremum points

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