Question

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]

Answer 1

(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).