The largest number in the sequence $1, 2^{\frac{1}{2}}, 3^{\frac{1}{3}}, 4^{\frac{1}{4}}, \ldots, n^{\frac{1}{n}}$ is

### Concept & Strategy This question touches upon a classic calculus optimization problem disguised as a sequence. The sequence represents the function $f(x) = x^{\frac{1}{x}}$. To find the maximum value of this sequence for integer values of $x$, we evaluate the function around its global maximum peak, which occurs at $x = e$ (Euler's number, approximately 2.718). Since $e$ falls between the integers 2 and 3, the maximum value in this integer sequence must occur at either $n = 2$ or $n = 3$. ### Step-by-Step Solution * **Given:** The sequence $n^{\frac{1}{n}}$ for natural numbers $n = 1, 2, 3, 4 \ldots$ * **Calculation / Deduction:** 1. By mathematical theory, the sequence increases until it hits $e$ and strictly decreases thereafter. Therefore, we only need to compare the integer values adjacent to $e$ (which are 2 and 3). 2. Let's manually compare $n=2$ and $n=3$: We need to determine which is larger: $2^{\frac{1}{2}}$ or $3^{\frac{1}{3}}$. 3. Take the LCM of the denominators of the powers (2 and 3), which is 6. 4. Raise both terms to the 6th power to remove fractional exponents: $$(2^{\frac{1}{2}})^6 = 2^3 = 8$$ $$(3^{\frac{1}{3}})^6 = 3^2 = 9$$ 5. Since $9 > 8$, we know that $3^{\frac{1}{3}} > 2^{\frac{1}{2}}$. 6. Because the sequence strictly decreases for all integers $n \geq 3$, $3^{\frac{1}{3}}$ must be the absolute largest number in the entire sequence. ### Exam Strategy & Shortcut **Fact Memorization:** For competitive exams like CAT/IIFT, the function $y = x^{\frac{1}{x}}$ is a standard curve to memorize. Its absolute maximum always occurs at $x = e \approx 2.718$. Since we are dealing with a sequence of integers, the maximum must be the integer closest to $2.718$, which intuitively points to 3. A quick check of 2 vs 3 confirms $3^{1/3}$ is indeed the peak. ### Common Pitfall Many students mistakenly believe that as the base $n$ increases, the overall value of $n^{\frac{1}{n}}$ must also continuously increase. They might guess options (c) or (d) thinking larger numbers yield larger results. Always remember that a growing denominator in an exponent ($1/n$) shrinks the value drastically, overpowering the growing base. ### Final Answer **Therefore, the correct answer is $3^{\frac{1}{3}}$.**