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

Last updated at Aug. 9, 2023 by Teachoo

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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Checking Assertion
ASSERTION (A): đ(đĽ) has a minimum at đĽ=1.
Finding minimum of đ(đ)
Differentiating w.r.t. x
đâ(đĽ) =(đĽâ1)^3 (đĽâ3)^2
Putting fâ(đ) = 0
(đĽâ1)^3 (đĽâ3)^2 = 0
Thus, either ă"(x â 1)" ă^2 = 0 and (đĽâ3)^2 = 0
â´ x = 1, 3
Checking maxima or minima at x = 1
â´ x = 1 is a point of minima
Hence, Assertion is true
Checking Reason
REASON (R): When đ/đđĽ(đ(đĽ))<0,âđĽâ(đââ, a ) and đ/đđĽ(đ(đĽ))>0,âđĽâ(đ,đ+â); where ^â˛ đ ' is an infinitesimally small positive quantity, then đ(đĽ) has a minimum at đ=đ, provided đ(đ) is continuous at đ=đ.
Here, reasoning is describing the 1st derivative test
fâ(x) < 0 for x â(đââ, a) i.e. fâ(x) < 0 on left of x = a
fâ(x) > 0 for x â(đ,đ+â) i.e. fâ(x) > 0 on right of x = a
Since when sign fâ(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)

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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