Differential Calculus
Differential Calculus
star_batch_jee_advanced_2025
Grade None
Question:
The function $f: (a, \infty) \to R$ where $R$ denotes the range corresponding to the given domain, with rule $f(x) = 2x^3 - 3x^2 + 6$ will have an inverse provided:
$a \geq 1$
$a \geq 0$
$a \leq 0$
$a \leq 1$
Step-by-Step Solution
Key Concept: Use the Sandwich theorem by finding tight lower and upper bounds that both converge to the same limit.
To find $\lim_{n\to\infty}f(n)$ where $f(n) = \sum_{k=1}^{n}\frac{k}{n^2+nk}$, we squeeze $f(n)$ between two bounds: $g(n) = \frac{n(n+1)}{2(n^2+2n)}$ (lower bound) and $h(n) = \frac{n(n+1)}{2(n^2+n+1)}$ (upper bound). Both bounds approach $\frac{1}{2}$ as $n\to\infty$. By the Sandwich theorem, $\lim_{n\to\infty}f(n) = \frac{1}{2}$.
Correct Answer: 3