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The negation of the statement
"For every real number \(x ; x^2+5\) is positive" is

A For every real number \(x ; x^2+5\) is not positive.
B For every real number \(x ; x^2+5\) is negative.
C There exists atleast one real number \(x\), such that \(x^2+5\) is not positive.
D There exists atleast one real number \(x\), such that \(x^2+5\) is positive.

Verified Solution

Answer & Solution

Correct Answer

(C) There exists atleast one real number \(x\), such that \(x^2+5\) is not positive.

Step-by-step Solution

Detailed explanation

The negation of statement "For every real number \(x ; x^2+5\) is positive" is there exists at least one real number \(x\) such that \(x^2+5\) is not positive.

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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The negation of the statement"For every real number \(x ; x^2+5\) is positive" is

Step-by-step Solution

The negation of the statement
"For every real number \(x ; x^2+5\) is positive" is