Let $f: \mathbf{R} \rightarrow(0, \infty)$ be a continuous decreasing function. Suppose $f(0), \dot{f}(1), \ldots, f(10)$ are in a geometric progression with common ratio $\frac{1}{5}$. In which of the following intervals does the value of $\int_0^{10} f(x) d x$ lie?

Let $f:(-1,2) \rightarrow \mathbf{R}$ be a differentiable function such that $f^{\prime}(x)=\frac{2}{x^2-5}$ and $f(0)=0$. Then in which of the following intervals does $f(1)$ lie?

For a natural number $n$, let $C_n$ be the curve in the $X Y$-plane given by $y=x^n$, where $0 \leq$ $x \leq 1$. Let $A_n$ denote the area of the region bounded between $C_n$ and $C_n+1$. Then the largest value of $A_n$ is

Let $f$ be a continuous function on $[0,1]$ and $F$ be its antiderivative. If $F(0)=1$ and $\int_0^1 f(x) d x=1$, then $F(1)$ is