# To prove the function is strictly decreasing (nominal interest rate convertible $p$ times a year)

How would one prove that the function $$f(p)=p[(1+i)^p-1]$$ is strictly decreasing? Here $$i>-1$$.

I've tried the usual approach, i.e. finding the derivative of $$f$$ and got a pretty terrible expression $$f'(p)=(1+i)^{1/p}(1-\frac 1p \log(1+i))-1$$ and I cannot find where the derivative is positive and where it's negative?

Any help would be appreciated.

If $$h$$ is increasing and $$g$$ is increasing, then $$h\cdot g$$ is increasing if for all $$x$$, $$h(x),g(x)> 0$$. Indeed, if $$x> y$$, we have $$h(x)\cdot g(x)>h(y)\cdot g(y)$$ by the increasing properties.
Similarly, if $$h$$ is increasing and positive and $$g$$ is decreasing and negative, then $$h\cdot g$$ is decreasing by looking at it as $$-(h\cdot -g)$$ since $$-g$$ would be positive and increasing.
Assume now for the moment that we are working on the domains $$(0,\infty)$$ and take $$h(p) = p$$ and $$g(p) = (1+i)^p - 1$$.
If $$i>0$$, $$g$$ is increasing and positive. Thus $$h\cdot g$$ will be increasing. If $$-1>i>0$$, $$g$$ is decreasing and negative and thus $$h\cdot g$$ will be decreasing. If $$i=0$$, $$g$$ is zero and thus $$h\cdot g$$ will be zero.