__Finding Extrema of a Function__

We want to find the extrema of the function

`> `
**f := sin(x)*exp(-2*x^2);**

`> `
**df := diff(f, x);**

`> `
**solve(%, x);**

Since maple can't solve this equation analytically, we ask for a numeric solution:

`> `
**fsolve(df, x);**

We don't trust this simple solution blindly, but try to get a qualitative overview by plotting the function:

`> `
**plot(f, x=-10..10);**

This doesn't seem to agree with the numeric result, so we take a closer look:

`> `
**plot(df, x=-6.5..-6);**

The high precision of maple calculations allows us to refine even more:

`> `
**plot(f, x=-6.4..-6.3);**

This at last confirms the former result. But we are more interested in the "main" peak between 0 and 1:

`> `
**fsolve(df, x, 0..1);**

And again we use a plot for a gross check of the numerics:

`> `
**plot(df, x=0..1);**

To get an overview of all solutions, we go back to the algebraic solution:

`> `
**minmax := solve(df, x);**

`> `
**sol := op(1,%);**

A direct plot of this equation doesn't help much:

`> `
**plot(sol, _Z=-10..10);**

The reason for the strange plot are of course the poles of the tan. A first improvement is to restrict the range values:

`> `
**plot(sol, _Z=-10..10, -100..100);**

Finally, the nasty lines at the poles disappear, if we plot only the continuous parts:

`> `
**plot(sol, _Z=-10..10, -100..100, discont=true, color=red);**

Only the region around 0 needs a closer look:

`> `
**plot(sol, _Z=-1..1);**