Let φ be the density of the $χ^2(n)$ distribution. Find the point at which $φ$ has its maximum.
I can't really find a way to solve this question! Any help is welcome.
EDIT: I am given this one hint, which I don't understand where it could be used (the algebra can be simplified by appropriate use of logarithms)
ok, Let's deal with the hint about logarithms.
The density of the chi-square distribution with $n$ degrees of freedom is $$ f(x) = \text{constant} \cdot x^{(n/2)-1} e^{-x/2}. $$ So you have $$ \log f(x) =\text{constant} + \left( \frac n 2 - 1\right) \log x - \frac x 2. $$ Since $\log$ is an increasing function, $f$ and $\log\circ f$ both reach their maximum values at the same point.
So you have $$ \frac d {dx} \log f(x) = \left( \frac n 2 -1 \right) \frac 1 x - \frac 1 2. $$ This should be positive if $0<x<\text{something}$ and negative if $x$ is larger than that.