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Statement-1 : For any real number x,(2x^...

Statement-1 : For any real number `x,(2x^(2))/(1+x^(4)) le1`
Statement-2: `A.M.geG.M.`

A

1

B

2

C

3

D

4

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The correct Answer is:
To prove the statement \( \frac{2x^2}{1+x^4} \leq 1 \) for any real number \( x \), we can follow these steps: ### Step 1: Start with the given inequality We need to prove: \[ \frac{2x^2}{1+x^4} \leq 1 \] ### Step 2: Rearranging the inequality To eliminate the fraction, we can multiply both sides by \( 1 + x^4 \) (which is always positive for real \( x \)): \[ 2x^2 \leq 1 + x^4 \] ### Step 3: Rearranging the terms Rearranging gives us: \[ x^4 - 2x^2 + 1 \geq 0 \] ### Step 4: Substituting \( y = x^2 \) Let \( y = x^2 \). Then the inequality becomes: \[ y^2 - 2y + 1 \geq 0 \] ### Step 5: Factoring the quadratic The expression can be factored as: \[ (y - 1)^2 \geq 0 \] ### Step 6: Analyzing the factored form Since the square of any real number is non-negative, we have: \[ (y - 1)^2 \geq 0 \] This is true for all \( y \) (and thus for all \( x \)). ### Conclusion Since \( (y - 1)^2 \geq 0 \) holds for all real \( y \), we conclude that: \[ \frac{2x^2}{1+x^4} \leq 1 \] is true for all real \( x \).
To prove the statement \( \frac{2x^2}{1+x^4} \leq 1 \) for any real number \( x \), we can follow these steps: ### Step 1: Start with the given inequality We need to prove: \[ \frac{2x^2}{1+x^4} \leq 1 \] ...
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