Match Function Properties with Natural Number and Point Values

To determine the correct match between List-I and List-II, we analyze each statement step-by-step:

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(P) Continuity of $f(x)$ on $[1, 2]$:

The function is given by $f(x) = \left[\dfrac{p(x)}{n}\right], \quad \text{where } p(x) = 10x^3 - 45x^2 + 60x + 35$ Differentiating $p(x)$ with respect to $x$: $p'(x) = 30x^2 - 90x + 60 = 30(x - 1)(x - 2)$ For $x \in (1, 2)$, $p'(x) < 0$, which means $p(x)$ is strictly decreasing on $[1, 2]$.

Evaluating $p(x)$ at the endpoints:

• At $x = 1$: $p(1) = 10(1)^3 - 45(1)^2 + 60(1) + 35 = 60$
• At $x = 2$: $p(2) = 10(8) - 45(4) + 60(2) + 35 = 55$

Hence, for $x \in [1, 2]$, the range of $p(x)$ is $[55, 60]$, and the argument of the greatest integer function is: $\dfrac{p(x)}{n} \in \left[\dfrac{55}{n}, \dfrac{60}{n}\right]$

For $f(x) = \left[\dfrac{p(x)}{n}\right]$ to be continuous on $[1, 2]$, the function must be constant on $[1, 2]$, which requires that no integer lies inside the interval $\left(\frac{55}{n}, \frac{60}{n}\right]$ and the interval must be contained within $[k, k+1)$ for some integer $k$: $k \le \dfrac{55}{n} \le \dfrac{60}{n} < k+1$

• For $n \le 8$:
  • If $n = 8$: $\left[\frac{55}{8}, \frac{60}{8}\right] = [6.875, 7.5]$, which contains the integer $7$, so $f(x)$ is discontinuous.
• For $n = 9$:
  • $\left[\frac{55}{9}, \frac{60}{9}\right] = [6.11\dots, 6.66\dots] \subset [6, 7)$
  • Here, $f(x) = 6$ is constant and continuous on $[1, 2]$.

Thus, the minimum natural number $n$ is $9$. $\textbf{(P)} \rightarrow \textbf{(2)}$

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(Q) Increasing property of $g(x)$ on $\mathbb{R}$:

The function is given by $g(x) = (2n^2 - 13n - 15)(x^3 + 3x)$ Taking the derivative with respect to $x$: $g'(x) = (2n^2 - 13n - 15)(3x^2 + 3) = 3(2n^2 - 13n - 15)(x^2 + 1)$ Since $x^2 + 1 > 0$ for all $x \in \mathbb{R}$, $g(x)$ is an increasing function on $\mathbb{R}$ if and only if: $2n^2 - 13n - 15 \ge 0 \implies (2n - 15)(n + 1) \ge 0$ Since $n \in \mathbb{N}$, $n + 1 > 0$, so: $2n - 15 \ge 0 \implies n \ge \dfrac{15}{2} = 7.5$ The smallest natural number satisfying this inequality is $n = 8$. $\textbf{(Q)} \rightarrow \textbf{(1)}$

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(R) Local minima of $h(x)$ at $x = 3$:

The function is $h(x) = (x^2 - 9)^n(x^2 + 2x + 3) = (x - 3)^n (x + 3)^n (x^2 + 2x + 3)$ Note that for all $x \in \mathbb{R}$, $x^2 + 2x + 3 = (x+1)^2 + 2 > 0$. In a neighborhood of $x = 3$, the factor $(x + 3)^n(x^2 + 2x + 3) > 0$.

• If $n$ is odd, $(x - 3)^n$ changes sign from negative to positive at $x = 3$, meaning $x = 3$ is a point of inflection.
• If $n$ is even, $(x - 3)^n \ge 0$ for all $x$ near $3$, and $h(3) = 0$, which implies $h(x) \ge h(3)$ in a neighborhood of $x = 3$. Hence, $x = 3$ is a point of local minimum.

The smallest natural number $n > 5$ that is even is $n = 6$. $\textbf{(R)} \rightarrow \textbf{(4)}$

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(S) Points of non-differentiability of $l(x)$:

The function is $l(x) = \sum_{k=0}^{4}\left(\sin |x - k| + \cos \left|x - k + \dfrac{1}{2}\right|\right)$

• Since the cosine function is even, $\cos |u| = \cos u$, so: $\cos \left|x - k + \dfrac{1}{2}\right| = \cos \left(x - k + \dfrac{1}{2}\right)$ which is infinitely differentiable for all $x \in \mathbb{R}$.
• For the sine term, the derivative of $\sin |u|$ at $u = 0$ is: $\lim_{u \to 0^+} \dfrac{\sin u - 0}{u} = 1 \quad \text{and} \quad \lim_{u \to 0^-} \dfrac{\sin(-u) - 0}{u} = -1$ Thus, $\sin |x - k|$ is non-differentiable at $x = k$.

For $k \in \{0, 1, 2, 3, 4\}$, the function $l(x)$ is not differentiable at exactly the $5$ points: $x_0 \in \{0, 1, 2, 3, 4\}$ Thus, the number of points of non-differentiability is $5$. $\textbf{(S)} \rightarrow \textbf{(3)}$

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Conclusion:

Matching the results:

• (P) $\rightarrow$ (2)
• (Q) $\rightarrow$ (1)
• (R) $\rightarrow$ (4)
• (S) $\rightarrow$ (3)

This corresponds to Option B.