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Class 12 Maths Chapter 6: Application of Derivatives Assertion-reason questions with answers

2 assertion-reason questions on Application of Derivatives, each with its answer and the reason, written from the NCERT chapter. For the CBSE board and JEE Main, which are set from the same NCERT chapter. Open a question's solution to check your working; then take the free test to find your next chapter. Each question gives an Assertion (A) and a Reason (R); choose how they stand to each other.

Question 1
Assertion (A): The function f(x) = x3 + x is strictly increasing on R. Reason (R): f′(x) = 3x2 + 1 > 0 for every real x.
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Answer: (a) Both A and R are true, and R is the correct explanation of A.

f′(x) = 3x2 + 1 is positive everywhere, and a positive derivative on an interval makes f increasing there (Theorem 1, strictly in this case). Both true; R explains A.

Question 2
Assertion (A): The function f(x) = (x − 1)3 + 2 has a local extremum at x = 1. Reason (R): For f(x) = (x − 1)3 + 2, f′(1) = 0.
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Answer: (d) A is false, but R is true.

f′(x) = 3(x − 1)2 vanishes at x = 1 but is positive on both sides, so f′ does not change sign: x = 1 is a point of inflexion, not an extremum. A is false; R is true (f′(c) = 0 does not by itself give an extremum).

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