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Calculus
Monotonicity, Cubic Equations, Roots of Equations
jee_adv_2026_mock_p2
Grade 12
Question:
Let f(x) = x^3 - 6x^2 + 9x + 2. Which of the following is/are correct about the equation f(x) = 0?
A. It has exactly one real root.
B. It has three distinct real roots.
C. The roots are in arithmetic progression.
D. The sum of roots is 6.
Step-by-Step Solution
Key Concept: Analyze the cubic function using derivatives to determine the number of real roots.
Step 1: f(x) = x^3 - 6x^2 + 9x + 2. Step 2: f'(x) = 3x^2 - 12x + 9 = 3(x-1)(x-3). Step 3: Local maximum at x=1: f(1) = 1 - 6 + 9 + 2 = 6 > 0. Local minimum at x=3: f(3) = 27 - 54 + 27 + 2 = 2 > 0. Step 4: Since both extrema are above the x-axis, the cubic crosses the x-axis only once. Step 5: Sum of roots = -(-6)/1 = 6. Thus A and D are true.
Correct Answer:A, D
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