# How can I find the nature of a turning point?

I have this function which has 4 turning points:

$$f(x)=18m^4 x+42m^3 x^2-54m^2 x^3-42 m x^4+36x^5$$ f(x)==18m^4 x+42m^3 x^2-54m^2 x^3-42 m x^4+36x^5

However, I can't seem to find out how to use the same function to find the nature of the points. Am I missing something? I'm quite new to Mathematica.

• What range of x and m are you interested in? What are the four turning points you found? Can you show the code you've used so far? Apr 11 at 20:45
• x is just an arbetrary variable, i apoligise for not specifying the value of m, it is 0.89003. The turning points are at {{x -> -0.693447}, {x -> -0.151408}, {x -> 0.514972}, {x -> 1.16058}} , the code i have currently is : "m = 0.89003 b[x_] := 18 m^4 x + 42 m^3 x^2 - 54 m^2 x^3 - 42 m x^4 + 36 x^5 / Solve[b'[x] == 0] // N / {{x -> -0.693447}, {x -> -0.151408}, {x -> 0.514972}, {x -> 1.16058}}" Apr 11 at 21:02

\$Version

(* "13.2.1 for Mac OS X ARM (64-bit) (January 27, 2023)" *)

Clear["Global*"]

f[m_, x_] := 18 m^4 x + 42 m^3 x^2 - 54 m^2 x^3 - 42 m x^4 + 36 x^5

max[m_] = Solve[{D[f[m, x], x] == 0, D[f[m, x], {x, 2}] < 0}, x, Reals];

min[m_] = Solve[{D[f[m, x], x] == 0, D[f[m, x], {x, 2}] > 0}, x, Reals];

With[{m = 0.89003},
Plot[f[m, x], {x, -1, 1.5},
Epilog -> {AbsolutePointSize,
Red, Tooltip[Point[{x, f[m, x]}], {x, f[m, x]}] /. max[m],
Blue, Tooltip[Point[{x, f[m, x]}], {x, f[m, x]}] /. min[m]},
PlotLegends -> Placed[
PointLegend[{Red, Blue}, {"maxima", "minima"}, LegendMarkerSize -> 13],
{0.75, 0.75}]]] • For generality you should probably add something like indet[m_] = Solve[{D[f[m, x], x] == 0, D[f[m, x], {x, 2}] == 0}, x, Reals]; for the critical points that may or may not be extremal. Apr 12 at 16:02
• @MichaelSeifert - I considered that, but my guess at what was meant by “turning points” did not include inflection points, i.e., where the curve is flat. Apr 12 at 16:06

If by "turning points" you mean the values of $$x$$ where $$\frac{\partial f(x)}{\partial x}=0$$, then the following will get you what you want:

f[x_] := 18 m^4 x + 42 m^3 x^2 - 54 m^2 x^3 - 42 m x^4 + 36 x^5
Reduce[D[f[x], x] == 0, x] // N
(* x == -0.779128 m || x == -0.170115 m || x == 0.578601 m || x == 1.30398 m *)
`