EXERCISE 7.3

EXERCISE 7.3 1. Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals. (i) f x [ , ] ∈− 1 1 (ii) f x x x tan , [ , ] ∈ 0 π (iii) f x x x log , [ , ] 2 7 2. 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 [ , ] 0 1 (ii) f x [ , ] ∈− 1 6 (iii) f x [ , ] 0 9 3. Explain why Lagrange’s mean value theorem is not applicable to the following functions in the respective intervals : (i) f x [ , ] ∈− 1 2 (ii) f x | |, [ , ] ∈− 1 3 4. 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 [ , ] ∈− 2 2 (ii) f x )( ), [ , ] 3 11 5. Show that the value in the conclusion of the mean value theorem for (i) f x ( ) = 1 on a closed interval of positive numbers [ , ] a b is ab (ii) f x Ax Bx ( ) = on any interval [ , ] a b is a 6. A race car driver is in kilometer stone 20. If his speed never exceeds 150 km/hr, what is the maximum kilometer stone he can reach in the next two hours. 7. Suppose that for a function f x ( ), ′ ≤ 1 for all 1 £ £ . Show that f 12th_Maths_Vol 2_EM_CH 7_Differential Calculus.indd 21 16-12-2022 16:25:45 22 8. Does there exist a differentiable function f x ( ) such that f = − and ′ 2 for all x . Justify your answer. 9. Show that there lies a point on the curve f x x x −≤ where tangent drawn is parallel to the x -axis. 10. Using mean value theorem prove that for, a > > , | | | | .