If f and g are differentiable function in 0 , 1 satisfying f 0 = 2 = g 1 , g 0 = 0 and f 1 = 6 , then for some c ∈ 0 , 1

Question

If f and g are differentiable function in 0 , 1 satisfying f 0 = 2 = g 1 , g 0 = 0 and f 1 = 6 , then for some c ∈ 0 , 1, 2 f ' c = g c, 2 f ' c = 3 g ' c, f ' c = g ' c, f ' c = 2 g ' c

MARKS App · Accepted Answer

Let h x = f x - 2 g x h 0 = f 0 - 2 g 0 h 0 = 2 - 2 0 ∵ f 0 = 2 , g 0 = 0 h 0 = 2 h 1 = f 1 - 2 g 1 h ( 1 ) = 6 - 2 ( 2 ) ∵ f 1 = 6 , g 1 = 2 h 1 = 2 h 0 = h 1 h ' x = f ' x - 2 g ' x By Rolle's theorem h ' c = 0 ∴ f ' c - 2 g ' c = 0 , f ' c = 2 g ' c

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