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CAT 2020 PYQsQuantitative Ability. Free, no login required.

Algebra

past year questions, CAT 2020

Minima & Maxima

past year questions, Algebra, CAT 2020

Medium

For real x,x, the maximum possible value of x1+x4\frac{x}{\sqrt{1+x^4}} is

Solution

✅ Correct Option: 1

Dividing by x on both the numerator and the denominator:

11+x4x2=11x2+x21x2+x2≥2(theorem)1x2+x2≥211x2+x2≤12\begin{aligned} \dfrac{1}{\sqrt{\tfrac{1+x^4}{x^2}}} &= \dfrac{1}{\sqrt{\tfrac{1}{x^2} + x^2}} \\ \dfrac{1}{x^2} + x^2 &\geq 2 \text{(theorem)}\\ \sqrt{\dfrac{1}{x^2} + x^2} &\geq \sqrt{2} \\ \dfrac{1}{\sqrt{\tfrac{1}{x^2} + x^2}} &\leq \dfrac{1}{\sqrt{2}} \end{aligned}

11+x4x2≤12\boxed{\dfrac{1}{\sqrt{\tfrac{1+x^4}{x^2}}} \le \dfrac{1}{\sqrt{2}}}

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