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The first part of this question involves an easy evaluation of the
integrals and then some simple algebra to compare the
results.

In the second part you will get a quadratic inequality in $\lambda$ of the form $$U + 2V\lambda + W\lambda^2 \geq 0.$$ Completing the square leads to a condition for this inequality to hold which gives the required result.

In the second part you will get a quadratic inequality in $\lambda$ of the form $$U + 2V\lambda + W\lambda^2 \geq 0.$$ Completing the square leads to a condition for this inequality to hold which gives the required result.