If a function $f$ is differentiable on $R$ such that $f^{\prime}(x) \leq 4$ for all $x \in R$; and if $f(2)=-6$ and $f(6)=8$,then the value of $f(4)$ belongs to the interval

The constant $c$ of Lagrange's mean value theorem for $f(x)=\cos x-\sin 2x$ in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ is

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...

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

The value of $c$ in the Lagrange's mean value theorem for the function $f(x) = x^{3} - 4x^{2} + 8x + 11$ on the interval $x \in [0, 1]$ is:

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