Assertion =\frac{2}{\pi} x \sin x+x^{3}, \) where \) Statement-1: =\frac{\pi}{2} \) has exactly one solution in \) and Reason Statement-2: \geq \mathbf{0} \) for all in \)

Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1

Maths · Applications of derivatives: Rate of change of quantities, monotonic-Increasing and decreasing functions, Maxima and minima of functions of one variable

Assertion =\frac{2}{\pi} x \sin x+x^{3}, \) where \) Statement-1: =\frac{\pi}{2}

Assertion \( f(x)=\frac{2}{\pi} x \sin x+x^{3}, \) where \( x \in \) \( \left[0, \frac{\pi}{2}\right] \) Statement-1: \( f(x)=\frac{\pi}{2} \) has exactly one solution in \( \boldsymbol{x} \in\left[\mathbf{0}, \frac{\boldsymbol{\pi}}{\mathbf{2}}\right] \) and Reason Statement-2: \( \boldsymbol{f}(\boldsymbol{x}) \geq \mathbf{0} \) for all \( \boldsymbol{x} \) in \( \left[0, \frac{\pi}{2}\right] \)

A. Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
B. Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Step-by-step solution

f(x) = (2/π)x sin x + x^3 is continuous and strictly increasing on [0, π/2] (since f'(x) > 0 for x > 0). f(0)=0 < π/2 < f(π/2)=1+π^3/8, so by the Intermediate Value Theorem, f(x)=π/2 has exactly one solution. Statement-2 (f(x) ≥ 0) is true because all terms are nonnegative on [0, π/2], but it alone does not imply the uniqueness of the solution; the strictly increasing property is the correct justification.

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Step-by-step solution

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