Im looking for different concave function between $[0,1]$ which is continuous and differentiable. The function value should be $0$ at $0$ and $1$ at $1$.
one such function is $2x – x^2$
[Math] Concave function between $0$ and $1$
functions
functions
Im looking for different concave function between $[0,1]$ which is continuous and differentiable. The function value should be $0$ at $0$ and $1$ at $1$.
one such function is $2x – x^2$
Best Answer
Take any continuous, negative function $f$. Integrating $f$ twice, you obtain $F$. Then consider $G(x) = F(x)+ax+b$, where $a$ and $b$ are chosen so as to ensure your boundary conditions. Then $G$ is a possible answer (by doing so, I believe you would find all such $C^2$ functions).
Take $f(x) = -x^2$. Then
$f_1(x) = \int_0^xf(y)dy = -\frac{x^3}3$
and
$F(x) = \int_0^xf_1(y)dy = -\frac{x^4}{12}.$
Accounting for the BC leads to
$G(x) = -\dfrac{x^4}{12}+\dfrac{13x}{12}$