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For every pair of continuous functions f, g: [0, 1] → R such that max{f(x): x ∈ [0, 1]} = max {g(x): x = [0, 1]}, the correct statement(s) is (are) 

(A) (f(c))2 + 3f (c) = (g(c))2 + 3g(c) for some c ∈ [0, 1]

(B) (f(c))2 + f(c)=(g(c))2 + 3g(c) for some c ∈ [0, 1]

(C) (f(c))2 + 3f (c) = (g(c))2 + g(c) for some c ∈ [0, 1] 

(D) (f(c))2 = (g(c))2 for some c ∈ [0, 1]

1 Answer

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Best answer

(A) (f(c))2 + 3f (c) = (g(c))2 + 3g(c) for some c ∈ [0, 1]

(D) (f(c))2 = (g(c))2 for some c ∈ [0, 1]

Let f (x) and g (x) achieve their maximum value at x1 and x2 respectively

h(x) = f(x) - g(x)

h(x1) = f(x1) - g(x1) ≥ 0

h (x2) = f(x2) - g(x2) ≤ 0

⇒ h (c) = 0 where c ∈ [0, 1] ⇒ f (c) = g(c).

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