Q249 / 270 · NDA 2025 (Phase 2) · Mathematics
Consider the following statements regarding the function $f(x) = \dfrac{1}{x - 5}$:
Statement-I: $f(x)$ is decreasing on the intervals $x < 5$ and $x > 5$.
Statement-II: $f'(x) > 0$ for all $x \neq 5$.
Which one of the following is correct in respect of the above statements?
Statement-I: $f(x)$ is decreasing on the intervals $x < 5$ and $x > 5$.
Statement-II: $f'(x) > 0$ for all $x \neq 5$.
Which one of the following is correct in respect of the above statements?
- ABoth Statement-I and Statement-II are correct and Statement-II explains Statement-I
- BBoth Statement-I and Statement-II are correct but Statement-II does not explain Statement-I
- Correct answerCStatement-I is correct but Statement-II is not correct
- DStatement-I is not correct but Statement-II is correct
Solution
Given the function $f(x) = \dfrac{1}{x - 5}$.
Differentiating with respect to $x$, we get:
$f'(x) = \dfrac{d}{dx}(x - 5)^{-1} = -1(x - 5)^{-2} = -\dfrac{1}{(x - 5)^2}$
For all $x \neq 5$, $(x - 5)^2 > 0$, which implies $f'(x) < 0$.
Since $f'(x) < 0$ for all $x \neq 5$, the function $f(x)$ is strictly decreasing on the interval $(-\infty, 5)$ and strictly decreasing on the interval $(5, \infty)$.
Therefore, Statement-I is correct.
Statement-II claims that $f'(x) > 0$ for all $x \neq 5$, which contradicts our finding that $f'(x) < 0$. Thus, Statement-II is not correct.
Hence, Statement-I is correct but Statement-II is not correct.
Answer: Statement-I is correct but Statement-II is not correct
Differentiating with respect to $x$, we get:
$f'(x) = \dfrac{d}{dx}(x - 5)^{-1} = -1(x - 5)^{-2} = -\dfrac{1}{(x - 5)^2}$
For all $x \neq 5$, $(x - 5)^2 > 0$, which implies $f'(x) < 0$.
Since $f'(x) < 0$ for all $x \neq 5$, the function $f(x)$ is strictly decreasing on the interval $(-\infty, 5)$ and strictly decreasing on the interval $(5, \infty)$.
Therefore, Statement-I is correct.
Statement-II claims that $f'(x) > 0$ for all $x \neq 5$, which contradicts our finding that $f'(x) < 0$. Thus, Statement-II is not correct.
Hence, Statement-I is correct but Statement-II is not correct.
Answer: Statement-I is correct but Statement-II is not correct
Hence, the correct answer is option: 'C'.