# Thread: Find minimum value for function to be invertible?

1. ## Find minimum value for function to be invertible?

Fin the minimum value of $\displaystyle a$ and $\displaystyle b$ for which $\displaystyle f(x) = x^x$ ; $\displaystyle f : [a,\infty] \rightarrow [b,\infty)$ to be an invertible function.

2. A function will be invertible as long as it does not have a "turning point"- as long as its derivative is not 0.

3. I know that for a fact. What do we do in the given question?

4. $\displaystyle f(x) = e^{x\log x}$

$\displaystyle f'(x) = e^{x\log x}(1/x+1)$

now solve for f'(x)=0

5. Originally Posted by Bruno J.
$\displaystyle f(x) = e^{x\log x}$

$\displaystyle f'(x) = e^{x\log x}(1/x+1)$

now solve for f'(x)=0
$\displaystyle f'(x)=x^x(1+\ln x)$

$\displaystyle f'(x)=0$

$\displaystyle x=\frac{1}{e}$

$\displaystyle f\left(\frac{1}{e}\right)=\frac{1}{e^{\frac{1}{e}} }$

Therefore, for f to be invertible,$\displaystyle f:\left[\frac{1}{e},\infty\right)\rightarrow\left[\frac{1}{e^{\frac{1}{e}}},\infty\right)$

Thus $\displaystyle a=\frac{1}{e}$ and $\displaystyle b=\frac{1}{e^{\frac{1}{e}}}$