[Assertion Reasoning Class 12] Let f(x) be a polynomial function of

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)

Question 19 [Assertion Reasoning] - CBSE Class 12 Sample Paper for 2024 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Let f(x) be a polynomial function of degree 6 such that d/dx(f(x))=(x-1)^3 (x-3)^2, then

ASSERTION (A) : f(x) has a minimum at x=1.

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 .

(a) Both A and R are true and R is the correct explanation of A.

(b) Both A and R are true but R is not the correct explanation of A.

(c) A is true but R is false.

(d) A is false but R is true.

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