How to find the number of real roots for this equation?

Question

I want to find the number of real roots that satisfy this equation:

f[g[x]] - h[f[x]] == (a)^2

Where:

f[x_] = x*Cos[x]
g[x_] = CubeRoot[-5 + (x)^2]
h[x_] = Abs[x + 2]

Where x is given as between the values of 0 and 5 i.e. [0,5]. And a is a parameter that i can change the value of.

What I have tried so far is to animate the function to just get a visual on the functions. I wrote this code:

Animate[Plot[{f[g[x]] - h[f[x]], (a)^2}, {x, 0, 5}], {a, 0, 1}]

I then wondered what might be the maximum value in this interval and I found it to be 0.511 when x = 2.4987. I used this code:

FindMaximum[{f[g[x]] - h[f[x]]}, {x, 2, 5}]

That value is given when a = 0.71499.

By a purely graphical analysis there are always two solutions to this equation at any one value of a when a is between the values of [0,0.71499). At a=0.71499 there is only one solution and beyond it there are no solutions. Now how do I calculate the total number of roots to this equations? I have tried the command CountRoots but it doesn't yield a result. Any other suggestions?

Answer 1

ContourPlot gives you an idea of the answer I think

f[x_] = x*Cos[x]
g[x_] = CubeRoot[-5 + (x)^2]
h[x_] = Abs[x + 2]

pic=ContourPlot[f[g[x]] - h[f[x]] == (a)^2, {x, 0, 5}, {a, -1, 1}, FrameLabel -> {x, a}]

The peak-values follow from

mini = NMinimize[{a, f[g[x]] - h[f[x]] == (a)^2, Element[a, Reals]}, {x, a}] 
(*{-0.715053, {x -> 2.49876, a -> -0.715053}}*)
maxi = NMaximize[{a, f[g[x]] - h[f[x]] == (a)^2,Element[a, Reals]}, {x, a}] 
(*{0.714933, {x -> 2.49863, a -> 0.714933}}*)

amin=a/.mini[[2]]; (*-0.715053*)
amax=a/.maxi[[2]]; (*0.714933*) 

As you can see from ContourPlot:

There exist two solutions if amin<a<amax There exists one solutions if a==amin or a==amax In all other cases there is no real solution

There exists no solution if `a>a/.maxi

Answer 2

You can try FindInstance:

FindInstance[f[g[x]] - h[f[x]] == a^2, {a, x}, Reals]

And

FindInstance[f[g[x]] - h[f[x]] == a^2, {a, x}, Complexes]

Each of these yielded one solution, both outside your outside domain. Neither found multiple solutions, but that is limited by the methods available to the function.

Answer 3

Look at the result of

f[x_] = x*Cos[x];g[x_] = CubeRoot[-5 + (x)^2] ;h[x_] = Abs[x + 2];
ContourPlot[{f[g[x]] - h[f[x]] == (a)^2}, {x, 0, 5}, {a, 0, 1}]

to this end.

Addition.

ContourPlot[{f[g[x]] - h[f[x]] == (a)^2}, {x, -50, 50}, {a, -5,5 },PlotPoints->100]