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Transcript

Checking Assertion ASSERTION (A): 𝑓(𝑥) has a minimum at 𝑥=1. Finding minimum of 𝒇(𝒙) Differentiating w.r.t. x 𝑓’(𝑥) =(𝑥−1)^3 (𝑥−3)^2 Putting f’(𝒙) = 0 (𝑥−1)^3 (𝑥−3)^2 = 0 Thus, either 〖"(x – 1)" 〗^2 = 0 and (𝑥−3)^2 = 0 ∴ x = 1, 3 Checking maxima or minima at x = 1 ∴ x = 1 is a point of minima Hence, Assertion is true Checking Reason REASON (R): When 𝑑/𝑑𝑥(𝑓(𝑥))<0,∀𝑥∈(𝑎−ℎ, a ) and 𝑑/𝑑𝑥(𝑓(𝑥))>0,∀𝑥∈(𝑎,𝑎+ℎ); where ^′ 𝒉 ' is an infinitesimally small positive quantity, then 𝑓(𝑥) has a minimum at 𝒙=𝒂, provided 𝒇(𝒙) is continuous at 𝒙=𝒂. Here, reasoning is describing the 1st derivative test f’(x) < 0 for x ∈(𝑎−ℎ, a) i.e. f’(x) < 0 on left of x = a f’(x) > 0 for x ∈(𝑎,𝑎+ℎ) i.e. f’(x) > 0 on right of x = a Since when sign f’(x) changes from negative to positive, it is Minima Hence, Reason is true Is Reason a Correct explanation for Assertion? Since we used the concept mentioned in Reasoning to check Assertion Therefore, Reasoning is a correct explanation for Assertion So, Assertion is true Reasoning is true And, Reasoning is a correct explanation for Assertion So, the correct answer is (a)

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo