MaplePrimes - answers and comments on Question, How to find the critical points and local maxima and minima by only using the 1st derivative?

Please check your code

toandhsp — Thu, 20 Sep 2012 19:53:30 Z

Please check this line df:=x->abs(1,fo(x))((3*x^2)-16*x+5)-0.5. I think

fo(x) must be ,f0(x).

You are on the right way

Markiyan Hirnyk — Thu, 20 Sep 2012 20:38:46 Z

You made a good start. Let us continue. The ?CriticalPoints command is preferable over the
the ?ExtermePoints command because the endpoints are not local extrema by definition. Therefore,
> CP := CriticalPoints(f(x), x = -1 .. 7);
[-(7/3)*cos((1/3)*arctan((3/278)*sqrt(3)*sqrt(1495)))+8/3-(7/3)*sin((1/3)*arctan((3/278)*
sqrt(3)*sqrt(1495)))*sqrt(3), 8/3-(1/6)*sqrt(202), -(7/3)*cos((1/3)*arctan((3/278)*sqrt(3)*
sqrt(1495)))+8/3+(7/3)*sin((1/3)*arctan((3/278)*sqrt(3)*sqrt(1495)))*sqrt(3), 8/3+(1/6)*
sqrt(190)]
> evalf(CP);

     [-0.4530286320, 0.297888267, 1.220859768, 4.964008126]
Next, a short code to saturate the maxima and minima.
> minima := []: maxima := []:
> for j to nops(CP) do if evalf(df(CP[j]-0.1e-3)) > 0 and evalf(df(CP[j]+0.1e-3)) < 0
then maxima := [op(maxima), CP[j]]
else if evalf(df(CP[j]-0.1e-3)) < 0 and evalf(df(CP[j]+0.1e-3)) > 0
then minima := [op(minima), CP[j]]
end if:
end if:
end do;
> maxima;
       [8/3-(1/6)*sqrt(202), 8/3+(1/6)*sqrt(190)]

> minima;
[-(7/3)*cos((1/3)*arctan((3/278)*sqrt(3)*sqrt(1495)))+8/3-(7/3)*sin((1/3)*arctan((3/278)*
sqrt(3)*sqrt(1495)))*sqrt(3), -(7/3)*cos((1/3)*arctan((3/278)*sqrt(3)*sqrt(1495)))+
8/3+(7/3)*sin((1/3)*arctan((3/278)*sqrt(3)*sqrt(1495)))*sqrt(3)]
> evalf(maxima);

                   [0.297888267, 4.964008126]
> evalf(minima);

                  [-0.4530286320, 1.220859768]

The plot
> plot(f, -1 .. 7)

confirms the obtained outputs.

extrema.mw

Yes you are right. It is ,f0(x) I typed

kverbarg — Thu, 20 Sep 2012 20:32:28 Z

Yes you are right. It is ,f0(x)

I typed it wrong here, but that is what maple printed out (for some reason i have to retype everything because i can't copy and paste)

what about the list of (x,y)?

kverbarg — Thu, 20 Sep 2012 21:22:12 Z

That helped a lot, thanks! But what about for the list? I need to put out 2 lists, one for the minimas and one for the maximas in (x,y) format

No problem

Markiyan Hirnyk — Thu, 20 Sep 2012 21:36:15 Z

@kverbarg It can be done as follows.

> ListTools:-Transpose([maxima, map(f, maxima)]);

I hope you will not have a problem with minima.

Thank you!!!

Cody — Thu, 20 Sep 2012 23:01:04 Z

@Markiyan Hirnyk Thank you!!!