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 $\displaystyle f(x)=x+x^2\sin\frac\pi x$ for $\displaystyle x\neq0$, $\displaystyle f(0)=0$ ? Clearly the function is differentiable at all non-zero $\displaystyle x$, whereas

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

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

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

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