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 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)= ≤ft 10x³-45x²+60x+35/n is continuous on the interval [1,2], is (1) 8 (Q) The minimum value of n for which g(x)= (2 n² - 13 n.-15) (x³+3 x) , x {R}, is an increasing function on{R}, is (2) 9 (R) The smallest natural numbernwhich is greater than 5, such thatx=3is a point of local minima ofh(x)= (x²-9)ⁿ (x²+2x+3), is (3) 5 (S) Number ofx₀ {R}such thatI(x)=_{k=0}^{4} ≤ft( \; |x-k|+ ≤ft| x-k+\;1/2 | ) , x {R}, is NOT differentiable atx₀, is (4) 6 (5) 10 [JEE Adv 2025 P1]

Question

Examsnet · Accepted Answer

Correct Answer is : (P) ⟶⟶ (2) (Q) ⟶⟶ (1) (R) ⟶⟶ (4) (S) ⟶⟶ (3)

	List-I		List-II
(P)	The minimum value of $n$ for which the function $f(x)= \left\lfloor \frac{10x^3-45x^2+60x+35}{n} \right\rfloor$ 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 \mathbb{R} $, is an increasing function on$ \mathbb{R} $, is </td> < td> (2) </td> < td> 9 </td> </tr> < tr> < td> (R) </td> < td> The smallest natural number$ 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}+2x+3) $, is </td> < td> (3) </td> < td> 5 </td> </tr> < tr> < td> (S) </td> < td> Number of$ x_{0} \in \mathbb{R} $such that$ I(x)=\sum\limits_{k=0}^{4} \left( \sin \; \|x-k\|+\cos \left\| x-k+\;\frac{1}{2} \right\| \right) , x \in \mathbb{R} $, is NOT differentiable at$ x_{0}$, is	(4)	6
		(5)	10

Limits, Continuity and Differentiability