Question

Question 1: Verify Rolle’s theorem for the function f (x) = x²+ 2x – 8, x ∈ [– 4, 2].

Class 12 - Math - Continuity and Differentiability

Answer 1

Function is continuous in[-4,2] as it is a polynomial function and polynomial function is always continuous.

(ii) f'(x) =2x+2 exists in [-4,2] , hence the given function is derivable.

(iii) f(-4) = 0 and f(2) = 0
thus f(-4) = f(x)



Conditions of Rolle’s theorem are satisfied, hence there exists, at least one €[-4,2] such that f'(c) = 0

=> 2c+2 = 0
c = -1

Answer 2

★ CONTINUITY AND DIFFERENTIABILITY ★

Rolle's theorem for f ( x ) = x² + 2x - 8 , x ∈ [ -4 , 2 ]

Continuous function in [ -4 , 2 ] aslike , it's a polynomial function , another point is , The function exists in a defined real domain range for [ -4 , 2 ] HENCE , it can obviously be differentiated , again ; f ( -4 ) = 16 - 8 - 8 = 0

Hence , f ( -4 ) = 0 and , similarly , f ( 2 ) = 0

proving to be the roots of the function ,

Hence , rolle's theorem is verified for the above function

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