1. ## Analysis differentiation

Hi, I can't seem to get anywhere with the following problems:

1. Find a differentiable function f: R->R such that f'(0)=1 but f is not monotonically increasing on any interval (0,d).

2. Show that there exists a differentiable function f: R->R such that (f(x))^5 + f(x) + x = 0 for all x. [Hint: if f exists and has an inverse g what equations must g satisfy?]

Any help would be much appreciated.

xxxxxx

2. 1. Find a differentiable function f: R->R such that f'(0)=1 but f is not monotonically increasing on any interval (0,d).
How about the function $f(x)=x+x^2\sin\frac\pi x$ for $x\neq0$, $f(0)=0$ ? Clearly the function is differentiable at all non-zero $x$, whereas

$\left|\frac{f(h)-f(0)}h-1\right|\leq|h|$ for $h\neq0$, making $f'(0)=1$.

Furthermore if $x\neq0$ then $f'(x)=1+2x\sin\frac\pi x-\pi\cos\frac\pi x$.

Thus $f'(1/n)=1-\pi(-1)^n$ for all positive integers $n$, showing that the function is not monotonic on any interval $(0,a)$.

Of course, the function $f'(x)$ cannot be continuous at $0$ if it is to meet the conditions of the problem, wouldn't you say?