Let ℝ denote the set of all real numbers. For a real number x , let...

Let

\mathbb{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 $f(x)=\left[\frac{10 x^3-45 x^2+60 x+35}{n}\right]$ is continuous on the interval $[1,2]$ ,is	(1)	8
(Q)	The minimum value of $n$ for which $g(x)=\left(2 n^2-13 n-15\right)\left(x^3+3 x\right),x \in \mathbb{R}$ ,is an increasing function on $\mathbb{R}$ ,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 $h(x)=\left(x^2-9\right)^{\mathrm{n}}\left(x^2+2 x+3\right)$ ,is	(3)	5
(S)	Number of $x_0 \in \mathbb{R}$ such that $l(x)=\sum_{k=0}^4\left(\sin \|x-k\|+\cos \left\|x-k+\frac{1}{2}\right\|\right),x \in \mathbb{R}$ ,is NOT differentiable at $x_0$ ,is	(4)	6
		(5)	10

Options

A $(\mathrm{P}) \rightarrow(1),(\mathrm{Q}) \rightarrow(3),(\mathrm{R}) \rightarrow(2),(\mathrm{S}) \rightarrow(5)$
B $(\mathrm{P}) \rightarrow(2),(\mathrm{Q}) \rightarrow(1),(\mathrm{R}) \rightarrow(4),(\mathrm{S}) \rightarrow(3)$
C $(\mathrm{P}) \rightarrow(5),(\mathrm{Q}) \rightarrow(1),(\mathrm{R}) \rightarrow(4),(\mathrm{S}) \rightarrow(3)$
D $(\mathrm{P}) \rightarrow(2),(\mathrm{Q}) \rightarrow(3),(\mathrm{R}) \rightarrow(1),(\mathrm{S}) \rightarrow(5)$

Correct answer

B. $(\mathrm{P}) \rightarrow(2),(\mathrm{Q}) \rightarrow(1),(\mathrm{R}) \rightarrow(4),(\mathrm{S}) \rightarrow(3)$

Step-by-step solution

\begin{aligned} & P(x)=10 x^3-45 x^2+60 x+35 \\ & P^{\prime}(x)=30(x-1)(x-2) \end{aligned}

\mathrm{P}(\mathrm{x})

decreases in

[1,2] \Rightarrow

Range of

\mathrm{P}(\mathrm{x})=[55,60] f(\mathrm{x})=\left[\frac{\mathrm{P}(\mathrm{x})}{\mathrm{n}}\right] \min

value of

\mathrm{n}=9

(Q) For

g(x)

to be increasing

2 n^2-13 n-15 \geq 0

(R)

\begin{aligned} & h(x)=\left(x^2-9\right)^n\left(x^2+2 x+3\right) \\ & h^{\prime}(x)=\left(x^2-9\right)^n(2 x+2)+\left(x^2+2 x+3\right) n\left(x^2-9\right)^{n-1} \cdot 2 x \\ & =\left(x^2-9\right)^{n-1}\left[2\left(x^2-9\right)(x+1)+2 n x\left(x^2+2 x+3\right)\right] \\ & =(x+3)^{n-1}(x-3)^{n-1} \cdot q(x)\end{aligned}

Derivative must change sign at

x=3

\begin{aligned} & \because \mathrm{n}-1=\text{ odd } \\ & \mathrm{n}=\text{ even } \\ & \mathrm{n}=6 \end{aligned}

(S)

\cos \left|\mathrm{x}-\mathrm{k}+\frac{1}{2}\right|

is differentiable everywhere

\Rightarrow \sin |\mathrm{x}-\mathrm{k}|

is NOT diff. at

\mathrm{k}=0,1,2,3,4$

Practice Application of Derivatives on Quantrex Academy →

More from Application of Derivatives

Consider the function f: (0, ∞) arrow (-∞, ∞) given by f(x) = x \, _e(x) - x + 1 . Then… JEE Advanced 2026 Let P be the point on the parabola y = x^2 such that the slope of the tangent to the… JEE Advanced 2026 Let f: ℝ arrow ℝ be a differentiable function such that f((x+y)/(3)) = (f(x) + f(y))/(3)… JEE Main 2026 _ 0 ≤ x ≤ π (16 ((x)/(2)) ^3((x)/(2))) is equal to: JEE Main 2026 Let f(x) be a polynomial of degree 5 , and have extrema at x = 1 and x = -1 . If _ x → 0… JEE Main 2026 The number of critical points of the function f(x) = cases |( x)/(x)|, & x ≠ 0 1, & x = 0… JEE Main 2026 Let f be a differentiable function satisfying f(x)=1-2 x+∫_ 0 ^ x e^ (x-t) f(t) dt, x ∈ R… JEE Main 2026 Consider the following three statements for the function f:(0, ∞) arrow ℝ defined by… JEE Main 2026 Let (2 α, α) be the largest interval in which the function f(t)= |t+1| t^ 2 , t 2 , is… JEE Main 2026 Let f(x)=x^ 2025 -x^ 2000 , x ∈[0,1] and the minimum value of the function f(x) in the… JEE Main 2026 Let f: ℝ → ℝ be a twice differentiable function such that f''(x) > 0 for all x ∈ ℝ and… JEE Main 2026 Let f: R arrow R be a twice differentiable function such that the quadratic equation f(x)… JEE Main 2026 Let ℝ denote the set of all real numbers. Define the function f: R arrow R by f(x)=\… JEE Advanced 2025 Let ℝ denote the set of all real numbers. Let f: ℝ arrow ℝ be defined by f(x)= cases (6… JEE Advanced 2025 Let the function f(x)=(x)/(3)+(3)/(x)+3, x ≠ 0 be strictly increasing in (-∞, α_1) U(α_2… JEE Main 2025 Let f: R arrow R be a polynomial function of degree four having extreme values at x=4 and… JEE Main 2025 Full Application of Derivatives list Application of Derivatives JEE Maths PYQs JEE Mathematics PYQs All JEE Advanced PYQs