Is this a proof? The problem states: Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function such that $\int_{x}^{1}f(t)dt\geq(1-x)^{2}$, for any $x\in\mathbb{R}$. Prove that $f(1)=0$.
I denoted by $F(x)=\int_{x}^{1}f(t)dt$ and I noticed that $F(1)=0$ and $F(x)\geq0=F(1)$ and now, $f(1)=\lim_{x\rightarrow1,x>1}\frac{F(x)-F(1)}{x-1}\geq0$ and $f(1)=\lim_{x\rightarrow1,x<1}\frac{F(x)-F(1)}{x-1}\leq0$, so $f(1)=0$. Is this correct?