Mean Value Theorems

The value of $c$ for which $f\left(x\right)$ holds Lagrange's mean value theorem is, where $f\left(x\right)=\sqrt{25-x^2}, x\in \left[1,5\right]$

Using Rolle's theorem, the equation $a_0{x}^n+a_1{x}^{n-1}+\dots +a_n=0$ has at least one root between $0$ and $1$, if

Let $f$ be differentiable for all $x$. If $f\left(1\right)=-2$ and $f^{\text{'}}\left(x\right)\ge 2$ $\forall x\in \left[1,6\right]$, then