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Calculus
Monotonicity, Inverse Functions, Cubic Equations
jee_adv_2026_mock_p1
Grade 12
Question:
Let f: R → R be defined as f(x) = x^3 - 3x^2 + 3x. Which of the following is/are correct?
A. f is one-one.
B. f is onto.
C. The equation f(x) = 0 has three distinct real roots.
D. f is strictly increasing on R.
Step-by-Step Solution
Key Concept: Rewrite f(x) = (x-1)^3 + 1 to analyze monotonicity and range.
Step 1: f(x) = x^3 - 3x^2 + 3x = (x-1)^3 + 1. Step 2: f'(x) = 3(x-1)^2 ≥ 0, with equality only at x=1, so strictly increasing. Step 3: Strictly increasing with range R => one-one and onto. Step 4: f(x) = 0 => (x-1)^3 = -1 => x = 0, only one real root. Thus A, B, D are true.
Correct Answer:A, B, D
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