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Which of the following inequalities is NOT fine?

  • Option 1)

    \int_{a}^{b}x\sin xdx\leqslant \sqrt{\int_{a}^{b}x^{2}dx\int_{a}^{b}sin^{2}xdx}

  • Option 2)

    \int_{a}^{b}\sin xdx\leqslant \sqrt{x\int_{a}^{b}sin^{2}xdx}

  • Option 3)

    \int_{a}^{b}x\cos xdx\leqslant \sqrt{x^{3}\int_{a}^{b}cos^{2}xdx}

  • Option 4)

    \int_{a}^{b} \ln x\cdot e^{x}\leqslant \sqrt{\int_{a}^{b}\ln^{2}xdx\times \int e^{2x}dx

 

Answers (1)

best_answer

As we learned

Schwarz - Bunyakovsky Inequality -

If f^{2}\left ( x \right )and g^{2}\left ( x \right )are integral on x\in \left [ a,b \right ]

then

\left | \int_{a}^{b} f(x)\, \, g(x)\, dx\right |\leq \sqrt{\int_{a}^{b}f^{2}(x)dx\int_{a}^{b}g^{2}\left ( x \right )dx}

-

 

 

Violates the Cauchy Schwarz-Bunyakovsky Inequality


Option 1)

\int_{a}^{b}x\sin xdx\leqslant \sqrt{\int_{a}^{b}x^{2}dx\int_{a}^{b}sin^{2}xdx}

Option 2)

\int_{a}^{b}\sin xdx\leqslant \sqrt{x\int_{a}^{b}sin^{2}xdx}

Option 3)

\int_{a}^{b}x\cos xdx\leqslant \sqrt{x^{3}\int_{a}^{b}cos^{2}xdx}

Option 4)

\int_{a}^{b} \ln x\cdot e^{x}\leqslant \sqrt{\int_{a}^{b}\ln^{2}xdx\times \int e^{2x}dx

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gaurav

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