Looking for function with specific properties

Question

I need a function $f$ that is arbitrarly times differentiable and which has integral $$\int _a^b f(x) dx $$ strictly positive (where $a$ and $b$ are fixed), and for all derivatives, we have $$f^{(n)}(b)=f^{(n)}(a)=0.$$ I tried with $\sin$ and $\cos$, but was unsuccessful. Does anybody know such a function?

Answer 1

Look at Wikipedia, bump function.

The main tool is the function $f(x)=0$ for $x\le 0$, and $f(x)=e^{-1/x^2}$ for $x\gt 0$. This is everywhere differentiable, and all of its derivatives at $0$ are $0$.

Then the function $f(1-x^2)$ does what you want, with $a=-1$ and $b=1$. We can shift $-1$ to $a$ and $1$ to $b$ using an appropriate linear function.