How to find x's (more than one) that minimize a function in r

Question

hx <- function(x){
  abs(16*x*exp(1)^(-4*x) - 0.55)
}

roots_gx <- optimize(hx, lower = 0, upper = 2)$minimum
roots_gx

One of the roots of g(x) is 0.78, struggling to find the other one

tried this code:

roots_gx <- optimize(hx, lower = 0, upper = 2)$minimum
roots_gx 

I was expecting two answers only got one

Answer 1

First graph the function and use the appropriate intervals to obtain the solutions:

curve(hx) # perhaps use l2 norm instead of l1 norm.

Now we could obtain the two solutions:

optimize(hx, lower = 0, upper = 0.5)$minimum
[1] 0.04040332
optimize(hx, lower = 0.5, upper = 1)$minimum
[1] 0.7807179

---

But this is a well defined function. Use the lambertW function with its 2 branches to obtain the two solutions:

LambertW::W(-0.55/4)/-4
[1] 0.04040481
LambertW::W(-0.55/4, -1)/-4
[1] 0.7807226

These solutions are the true solutions as compared to the optimization above