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

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

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

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

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

Solution:

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)
The correct options are therefore, A, D