If c is a point at which Rolle's theorem holds for the function,
f(x) = $${\log _e}\left( {{{{x^2} + \alpha } \over {7x}}} \right)$$ in the interval [3, 4], where a $$ \in $$ R, then ƒ''(c) is equal to

Let ƒ(x) be a polynomial of degree 5 such that x = ±1 are its critical points.

If $$\mathop {\lim }\limits_{x \to 0} \left( {2 + {{f\left( x \right)} \over {{x^3}}}} \right) = 4$$, then which one of the following is not true?

The value of c in the Lagrange's mean value theorem for the function
ƒ(x) = x³ - 4x² + 8x + 11, when x $$ \in $$ [0, 1] is:

Let the function, ƒ:[-7, 0]$$ \to $$R be continuous on [-7,0] and differentiable on (-7, 0). If ƒ(-7) = - 3 and ƒ'(x) $$ \le $$ 2, for all x $$ \in $$ (-7,0), then for all such functions ƒ, ƒ(-1) + ƒ(0) lies in the interval: