The value of $c$ for Lagrange's Mean Value Theorem for $f(x) = \sqrt{25-x^2}$ on the interval $[1, 5]$ is

Verify the Mean Value Theorem for the function $f(x) = x^{2}$ in the interval $[2, 4]$.

If $a, b, c$ are real numbers such that $a + b + c = 0$,then the quadratic equation $3ax^2 + 2bx + c = 0$ has

The constant $c$ of Rolle's theorem for the function $f(x)=(x-1)^3(x-2)^5$ in the interval $[1, 2]$ is:

Let $a, b, c$ be real numbers such that $2a + 3b + 6c = 0$ and $g(x) = ax^2 + bx + c = 0$ has at least one root in the interval $(1, 2)$. If a function $f: [1, 2] \rightarrow \mathbb{R}$ for which Rolle's Theorem holds is such that $f(x)$ is a primitive of $g(x)$,then $f(x) = $

If the function $f(x) = x(x+3)e^{-x/2}$ satisfies all the conditions of Rolle's theorem in $[-3, 0]$, then a root of $f'(x) = 0$ is