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Take the function

$$f(x) = ax^{2} + bx^{4} - c \cos(x/d),$$

where $a$, $b$, $c$ and $d$ are arbitrary parameters.

For some given choice of the parameters, how do you find the number of local minima of the function and the location of the minima?

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closed as off-topic by rhermans, yohbs, happy fish, glS, MarcoB May 12 '17 at 13:15

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  • $\begingroup$ Related: (5575), (92698) $\endgroup$ – Michael E2 May 11 '17 at 22:23
  • $\begingroup$ Your question may (should?) be put on-hold or migrated because it seems to be off-topic, i.e. its about mathematics and not Mathematica. If that's not your intention, please edit your question to make it explicitly about Wolfram Mathematica programming. Include a formatted minimum example of the code you are working on $\endgroup$ – rhermans May 11 '17 at 22:25
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a = 0.01;
b = 0.0001;
c = 10;
d = 3;
sols = NSolve[
    D[a x^2 + b x^4 - c Cos[x/d], x] == 0 && 
    D[a x^2 + b x^4 - c Cos[x/d], {x, 2}] > 0, x, Reals];
{Length[sols], sols}

$\{3,\{\{x\to -16.6935\},\{x\to 0.\},\{x\to 16.6935\}\}\}$

The second derivative condition ensures a minimum.

Plot[a x^2 + b x^4 - c Cos[x/d], {x, -20, 20}, 
 Epilog -> {Red, PointSize[0.02], 
   Point@Transpose[{(x /. sols), {0, 0, 0}}]}]

enter image description here

Notice that your function is symmetric with respect to the interchange $x \leftrightarrow -x$, so you have an odd number of solutions, i.e., one at $x = 0$ and an even number of symmetric solutions.

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  • $\begingroup$ Since the OP asked about "minima" the solutions are sols = NSolve[{f'[x] == 0, f''[x] > 0}, x, Reals] which gives three minima. $\endgroup$ – Bob Hanlon May 11 '17 at 23:14
  • $\begingroup$ Ah yes.... I'll edit my solution when I get a chance. Thanks. $\endgroup$ – David G. Stork May 11 '17 at 23:23
  • $\begingroup$ @BobHanlon, what if the 2nd derivative is also zero vs h.o.t? $\endgroup$ – alancalvitti May 11 '17 at 23:26
  • $\begingroup$ @alancalvitti - I do not know what you mean by "h.o.t." If the second derivative is zero it could be an inflection point rather than a minima or maxima. $\endgroup$ – Bob Hanlon May 11 '17 at 23:41
  • $\begingroup$ @BobHanlon, higher order terms. $\endgroup$ – alancalvitti May 11 '17 at 23:45

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