Mean Value Theorem

Does there exist a differentiable function f(x) such that f(0) = – 1, f(2) = 4 and f(x) ≤ 2 for all x. Justify your answer

Explain why Lagrange’s mean value theorem is not applicable to the following functions in the respective intervals:

`f(x) = (x + 1)/x, x ∈ [-1, 2]`

Using the Lagrange’s mean value theorem determine the values of x at which the tangent is parallel to the secant line at the end points of the given interval:

`f(x) = x^3 - 3x + 2, x ∈ [-2, 2]`

A race car driver is kilometer stone 20. If his speed never exceeds 150 km/hr, what is the maximum kilometer he can reach in the next two hours

Show that the value in the conclusion of the mean value theorem for `f(x) = "A"x^2 + "B"x + "C"` on any interval [a, b] is `("a" + "b")/2`

Show that there lies a point on the curve `f(x) = x(x + 3)e^(pi/2), -3 ≤ x ≤ 0` where tangent drawn is parallel to the x-axis

Choose the correct alternative:

The number given by the Rolle’s theorem for the function x³ – 3x², x ∈ [0, 3] is

Choose the correct alternative:

The number given by the Mean value theorem for the function `1/x`, x ∈ [1, 9] is