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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J. M.'s missing motivation
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nightmarish
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Related: (5575), (92698) – Michael E2 May 11 '17 at 22:23
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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 – rhermans May 11 '17 at 22:25
1 Answers
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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}}]}]
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.
David G. Stork
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Since the OP asked about "minima" the solutions are
sols = NSolve[{f'[x] == 0, f''[x] > 0}, x, Reals]which gives three minima. – Bob Hanlon May 11 '17 at 23:14 -
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@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. – Bob Hanlon May 11 '17 at 23:41
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