Let f(x) = ax^3 + bx^2 + cx + 1 have extrema at x = , such that < 0 and f( ) f( ) < 0. Then the equation f(x) = 0 has

JEE Advanced Mathematics question

Mathematics Application of Derivatives JEE Advanced

Let have extrema at such that and . Then the equation has

(A)three equal real roots
(B)three distinct real rootsCorrect
(C)one positive root if and Correct
(D)one negative root if and Correct

Answer: (B),(C),(D)

Solution

is a cubic (, since it has two extrema) with critical points . On each of the three intervals , , , is strictly monotonic. - **(b) ✔** Since and have opposite signs, has exactly one root $

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Question ID VK-109415