Let's say we'd like to find the critical points of the function $f(x)=\sqrt{x-x^2}$. Finding out where the derivative is 0 is straightforward with Reduce
:
f[x_] := Sqrt[x - x^2]
f'[x] == 0
Reduce[%]
which yields:
(1 - 2 x)/(2 Sqrt[x - x^2]) == 0
x == 1/2
To find out where the real values of the derivative do not exist, I look for values of $x$ that make the denominator 0:
Reduce[2 Sqrt[x - x^2] == 0]
which yields:
x == 0 || x == 1
To do this, I manually extracted the denominator expression from the derivative and ran it through Reduce
.
My question is, is there a better way to ask Mathematica for values of $x$ where an expression doesn't have a real value? E.g. in this case, I'd like to pass in the expression for the derivative $\frac{1-2x}{2\sqrt{x-x^2}}$, and have it report that at 0 and 1, the real value doesn't exist.