Let R denote the set of all real numbers. For a real number x , let [ x ] denote the greatest integer less than or equal to x . Let n denote a natural number. Match each entry in List-I to the correct

Question

Let R denote the set of all real numbers. For a real number x , let [ x ] denote the greatest integer less than or equal to x . Let n denote a natural number. Match each entry in List-I to the correct entry in List-II and choose the correct option. LIST - I LIST - II (P) The minimum value of n for which the function f ( x ) = [ n 10 x 3 − 45 x 2 + 60 x + 35 ​ ] is continuous on the interval [ 1 , 2 ] ,is (1) 8 (Q) The minimum value of n for which g ( x ) = ( 2 n 2 − 13 n − 15 ) ( x 3 + 3 x ) , x ∈ R ,is an increasing function on R ,is (2) 9 (R) The smallest natural number n which is greater than 5,such that x = 3 is a point of local minima of h ( x ) = ( x 2 − 9 ) n ( x 2 + 2 x + 3 ) ,is (3) 5 (S) Number of x 0 ​ ∈ R such that l ( x ) = ∑ k = 0 4 ​ ( sin ∣ x − k ∣ + cos ​ x − k + 2 1 ​ ​ ) , x ∈ R ,is NOT differentiable at x 0 ​ ,is (4) 6 (5) 10

TestHub · Accepted Answer

​ P ( x ) = 10 x 3 − 45 x 2 + 60 x + 35 P ′ ( x ) = 30 ( x − 1 ) ( x − 2 ) ​ P ( x ) decreases in [ 1 , 2 ] ⇒ Range of P ( x ) = [ 55 , 60 ] f ( x ) = [ n P ( x ) ​ ] min value of n = 9 (Q) For g ( x ) to be increasing 2 n 2 − 13 n − 15 ≥ 0 (R) ​ h ( x ) = ( x 2 − 9 ) n ( x 2 + 2 x + 3 ) h ′ ( x ) = ( x 2 − 9 ) n ( 2 x + 2 ) + ( x 2 + 2 x + 3 ) n ( x 2 − 9 ) n − 1 ⋅ 2 x = ( x 2 − 9 ) n − 1 [ 2 ( x 2 − 9 ) ( x + 1 ) + 2 n x ( x 2 + 2 x + 3 ) ] = ( x + 3 ) n − 1 ( x − 3 ) n − 1 ⋅ q ( x ) ​ Derivative must change sign at x = 3 ​ ∵ n − 1 = odd n = even n = 6 ​ (S) cos ​ x − k + 2 1 ​ ​ is differentiable everywhere ⇒ sin ∣ x − k ∣ is NOT diff. at k = 0 , 1 , 2 , 3 , 4

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Doubts & Discussion

	LIST - I		LIST - II
(P)	The minimum value of $n$ for which the function $f (x) = [\frac{10 x ^{3} - 45 x ^{2} + 60 x + 35}{n}]$ is continuous on the interval $[1, 2]$ ,is	(1)	8
(Q)	The minimum value of $n$ for which $g (x) = (2 n^{2} - 13 n - 15) (x^{3} + 3 x), x \in R$ ,is an increasing function on $R$ ,is	(2)	9
(R)	The smallest natural number $n$ which is greater than 5,such that $x = 3$ is a point of local minima of $h (x) = (x^{2} - 9)^{n} (x^{2} + 2 x + 3)$ ,is	(3)	5
(S)	Number of $x_{0} \in R$ such that $l (x) = \sum_{k = 0}^{4} (sin ∣ x - k ∣ + cos x - k + \frac{1}{2}), x \in R$ ,is NOT differentiable at $x_{0}$ ,is	(4)	6
		(5)	10