CAT 2020 Quant Question [Slot 2] with Solution 10

Question

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

  1. \(\frac{1}{{\sqrt 3 }}\)
  2. 1
  3. \(\frac{1}{{\sqrt 2 }}\)
  4. \(\frac{1}{2}\)
Option: 3
Solution:

\(\frac{x}{{\sqrt {1 + {x^4}} }} = \frac{1}{{\sqrt {\frac{1}{{{x^2}}} + {x^2}} }}\)

\({x^2} + \frac{1}{{{x^2}}} \ge 2\)

Hence the maximum value of \(\frac{1}{{\sqrt {\frac{1}{{{x^2}}} + {x^2}} }}\) is \(\frac{1}{{\sqrt 2 }}\)

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