EXERCISE 7.3

EXERCISE . . Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals. (i) f x [ , ] ∈− (ii) f x x x tan , [ , ] ∈ π (iii) f x x x log , [ , ] . Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x -axis for the following functions : (i) f x x x [ , ] (ii) f x [ , ] ∈− (iii) f x [ , ] . Explain why Lagrange’s mean value theorem is not applicable to the following functions in the respective intervals : (i) f x [ , ] ∈− (ii) f x | |, [ , ] ∈− . 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: (i) f x [ , ] ∈− (ii) f x )( ), [ , ] . Show that the value in the conclusion of the mean value theorem for (i) f x ( ) = on a closed interval of positive numbers [ , ] a b is ab (ii) f x Ax Bx ( ) = on any interval [ , ] a b is a . A race car driver is in kilometer stone . If his speed never exceeds km/hr, what is the maximum kilometer stone he can reach in the next two hours. . Suppose that for a function f x ( ), ′ ≤ for all £ £ . Show that f . Does there exist a differentiable function f x ( ) such that f = − and ′ for all x . Justify your answer. . Show that there lies a point on the curve f x x x −≤ where tangent drawn is parallel to the x -axis. . Using mean value theorem prove that for, a > > , | | | | .