For the function $f(x)=(x-1)(x-2)$ defined on $\left[0, \frac{1}{2}\right]$,the value of $c$ satisfying Lagrange's 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

If $f(x) = x^\alpha \log x$ and $f(0) = 0$,then the value of $\alpha$ for which Rolle's theorem can be applied in $[0, 1]$ is

Let $f$ be a function that is derivable on the interval $[0, 1]$. Then, which of the following statements is true?

Find the value of $p$ and $q$ if the function $f(t) = t^3 - 6t^2 + pt + q$ defined on $[1, 3]$ satisfies Rolle's theorem for $c = \frac{2\sqrt{3} + 1}{\sqrt{3}}$.

Let $a > 0$ and $f$ be continuous in $[-a, a]$. Suppose that $f'(x)$ exists and $f'(x) \le 1$ for all $x \in (-a, a)$. If $f(a) = a$ and $f(-a) = -a$,then $f(0)$ is: