It get's kind of stuck if you choose unsuitable initial conditions. You can see that when you evaluate:

FindMaximum[{10^(-6) x^2, 0 <= x <= 10}, {x, #}, WorkingPrecision -> 30] & /@ Range[0, 10]

which basically uses different initial guesses in the interval. It is as I said above, Maximize is better. But if you absolutely want FindMaximum this does work:

Max[x /. (Quiet[FindMaximum[{10^(-6) x^2, 0 <= x <= 10}, {x, #}, WorkingPrecision -> 30] & /@ Range[0, 10]])[[All, 2]]]

which is really, really inefficient and slow. The Quiet is to ignore all (important) warnings that the method does not converge in the prescribed number of iterations.

M.

Maximize or NMaximize is more suitable for that problem. FindMaximum tries to find a local maximum an uses numarical evaluations. Also see the "Possible Issues" part of the documentation, espacially the part about numerical bumpyness and constraint regions.

I guess that this is a numerical artefact. If you set the precision higher you get a warning:

FindMaximum[{SetPrecision[10^(-6 ) x^2, Infinity]}, {x, 0}]

gives

FindMaximum::fmgz: Encountered a gradient that is effectively zero. The result returned may not be a maximum; it may be a minimum or a saddle point. >>

Maximize does work

Maximize[{10^(-6 ) x^2, 0 <= x <= 10}, x]

Cheers, Marco