Question:
Statement-1: The function \(x^2(e^x+e^{-x})\) is increasing for all \(x \gt 0\).
Statement-2: The functions \(x^2e^x\) and \(x^2e^{-x}\) are increasing for all \(x \gt 0\) and the sum of two increasing functions in any interval \((a,b)\) is an increasing function in \((a,b)\).
Statement-1: The function \(x^2(e^x+e^{-x})\) is increasing for all \(x \gt 0\).
Statement-2: The functions \(x^2e^x\) and \(x^2e^{-x}\) are increasing for all \(x \gt 0\) and the sum of two increasing functions in any interval \((a,b)\) is an increasing function in \((a,b)\).
Statement-2: The functions \(x^2e^x\) and \(x^2e^{-x}\) are increasing for all \(x \gt 0\) and the sum of two increasing functions in any interval \((a,b)\) is an increasing function in \((a,b)\).
Show Hint
Check the sign of the derivative of each function for \(x \gt 0\). Note that \(x^2e^{-x}\) has derivative \(xe^{-x}(2-x)\).
Updated On: Oct 31, 2026
- Statement-1 is false; Statement-2 is true.
- Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- Statement-1 is true; Statement-2 is false.
- Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
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The Correct Option is C
Solution and Explanation
Statement-1: let \(f(x)=x^2(e^x+e^{-x})\).
\(f'(x)=2x(e^x+e^{-x})+x^2(e^x-e^{-x})\).
For \(x \gt 0\): \(e^x+e^{-x} \gt 0\), \(e^x-e^{-x} \gt 0\), so every term is positive and \(f'(x) \gt 0\). Hence Statement-1 is true.
Statement-2: \(g(x)=x^2e^{-x}\) gives \(g'(x)=2xe^{-x}-x^2e^{-x}=xe^{-x}(2-x)\).
For \(x \gt 2\), \(g'(x) \lt 0\), so \(x^2e^{-x}\) is decreasing for \(x \gt 2\); it is NOT increasing for all \(x \gt 0\). Hence Statement-2 is false.
Answer: (3) Statement-1 is true; Statement-2 is false.
\(f'(x)=2x(e^x+e^{-x})+x^2(e^x-e^{-x})\).
For \(x \gt 0\): \(e^x+e^{-x} \gt 0\), \(e^x-e^{-x} \gt 0\), so every term is positive and \(f'(x) \gt 0\). Hence Statement-1 is true.
Statement-2: \(g(x)=x^2e^{-x}\) gives \(g'(x)=2xe^{-x}-x^2e^{-x}=xe^{-x}(2-x)\).
For \(x \gt 2\), \(g'(x) \lt 0\), so \(x^2e^{-x}\) is decreasing for \(x \gt 2\); it is NOT increasing for all \(x \gt 0\). Hence Statement-2 is false.
Answer: (3) Statement-1 is true; Statement-2 is false.
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