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Functions
Functional Equations, Cauchy's Equation
jee_adv_2026_mock_p2
Grade 12
Question:
Let f: R → R be a continuous function such that f(x+y) = f(x) + f(y) + 2xy for all x, y ∈ R. Which of the following is/are correct?
A. f(0) = 0.
B. f(x) = x^2 + cx for some constant c.
C. f is differentiable.
D. f is strictly increasing for all x.
Step-by-Step Solution
Key Concept: Use Cauchy's functional equation after subtracting x^2.
Step 1: Putting x = y = 0 gives f(0) = 2f(0) => f(0) = 0. Step 2: Let g(x) = f(x) - x^2. Then g(x+y) = g(x) + g(y). Step 3: With continuity, g satisfies Cauchy's equation, so g(x) = cx for some c. Step 4: Thus f(x) = x^2 + cx, which is differentiable. Step 5: It is not necessarily strictly increasing (depends on c). So A, B, C are true, D is false.
Correct Answer:A, B, C
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