Let $f(x)$ be differentiable on $[1, 6]$ and $f(1) = -2$. If $f(x)$ has only one root in $(1, 6)$, then there exists $c \in (1, 6)$ such that:

Let $f(x) = 8x^3 - 6x^2 - 2x + 1,$ then

Which of the following functions satisfies the conditions of Rolle's theorem on the given interval?

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

For a polynomial $g(x)$ with real coefficients,let $m_g$ denote the number of distinct real roots of $g(x)$. Suppose $S$ is the set of polynomials with real coefficients defined by $S = \{(x^2-1)^2(a_0+a_1x+a_2x^2+a_3x^3) : a_0, a_1, a_2, a_3 \in \mathbb{R}\}$. For a polynomial $f$,let $f'$ and $f''$ denote its first and second order derivatives,respectively. Then the minimum possible value of $(m_f + m_{f'})$,where $f \in S$,is

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