# How to solve this equation? or get the numbers of solutions of the function?

I have a function like this.. $$f[k]=\int_0^\infty Exp[-ax]x(Cos[bx])^{2}BesslJ[0,kx]dx$$ I want to know how many local maximuns this function has.. So I try to get the derivative of f[k] $$g[k]=D[f[k],k];$$ Then the result from mathematica is $$g[k]=-6ka^{-4}(1+k^{2}/a^{2})^{-5/2}-3k(a-2ib)^{-4}(1+k^{2}/(a-2ib)^{2})^{-5/2}-3k(a+2ib)^{-4}(1+k^{2}/(a+2ib)^{2})^{-5/2},.$$

I need to get the results $$g[k]==0,k=??$$ Because in the function there is a & b,I use Solve..But it can't get the answer..Actually just the number of the solutions is necessary for me.. I am a beginner of the Mathematica..TKS in advance.

• Please post the code for f[x]. LaTeX is fine for typesetting, but it's better to have actual code. – DumpsterDoofus Feb 4 '15 at 0:51
• TKS for your suggestion.. – Cici Feb 4 '15 at 0:57
• One solution is x = 0. I believe that there are two more. – bbgodfrey Feb 4 '15 at 1:07
• YES..x=0 is one of the answer...But I need the exact numbers of this function...Any solutions? – Cici Feb 4 '15 at 1:09
• Are a and b both real with a > 0? – bbgodfrey Feb 4 '15 at 1:54

g[k]=0 is not easily simplified. So, I plotted g (with k and b both scaled to a):

Manipulate[ Plot[g /. {a -> 1, b -> b0}, {k, 0, 20}, PlotRange -> {-.01, .01}],
{{b0, 1}, 0, 10, Appearance -> "Labeled"}]


A sample curve, for b = 6.12, is

Manifestly, there are three zeros, one at k = 0 and the others at positions dependent on b. (The curve does not cross the axis at large k.)

It also is instructive to plot f itself (again for b = 6.12)
Thus, f has 2 maxima and one minimum.
• Actually a->1 is a renormalization without loss of generality. Glad to be of help. – bbgodfrey Feb 4 '15 at 3:17