If $a + b + c = 0$,then how many roots does the equation $3ax^2 + 2bx + c = 0$ have in the interval $(0, 1)$?

The number of values of $C$ that satisfy the conclusion of Rolle's theorem for the function $f(x) = \sin(2 \pi x)$ on the interval $x \in [-1, 1]$ is:

If $f:[a, b] \rightarrow [c, d]$ is a continuous and strictly increasing function,then $\frac{d-c}{b-a}$ is

If $f(x) = \cos x$ for $0 \le x \le \frac{\pi}{2}$,then the real number $c$ of the Mean Value Theorem is:

If the function $f(x) = x^3 - 6ax^2 + 5x$ satisfies the conditions of Lagrange's mean value theorem for the interval $[1, 2]$ and the tangent to the curve $y = f(x)$ at $x = \frac{7}{4}$ is parallel to the chord that joins the points of intersection of the curve with the ordinates $x = 1$ and $x = 2$,then the value of $a$ is

Verify Rolle's theorem for the function $y=x^{2}+2$ on the interval $[-2, 2]$.