The position of a moving car at time $t$ is given by $f(t) = at^{2} + bt + c, t > 0,$ where $a, b,$ and $c$ are real numbers greater than $1.$ Then the average speed of the car over the time interval $[t_{1}, t_{2}]$ is attained at the point

The function $f(x) = x^3 - 6x^2 + ax + b$ satisfies the conditions of Rolle's theorem in $[1, 3]$. Then the values of $a$ and $b$ are respectively

Let $f(x)$ be a differentiable function in $[0, 2]$,$f(0) = 0$ and $f'(x) \le \frac{1}{2}$ for all $x \in [0, 2]$. Then:

Let $f(x)$ satisfy all the conditions of the Mean Value Theorem in $[0, 2]$. If $f(0) = 0$ and $|f'(x)| \le \frac{1}{2}$ for all $x$ in $[0, 2]$,then:

Rolle's theorem holds for a monic quadratic polynomial $f(x)$ on the interval $[\alpha, \alpha + 3]$ where $f(\alpha) = 0$. Similarly, $g(x) = f(x) + 2$ also follows Rolle's theorem on the interval $[\beta, 3]$ where $g(3) = 0$, such that the value of $c$ (where $f'(c) = g'(c) = 0$) is the same for both $f(x)$ and $g(x)$. Then the value of $(f \circ g)(\alpha)$ is...

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