QJEEM11Mac529: JEE Main 2011 Mathematics - Class 12 JEEM11Mac529 Solution: For define Then has

Question

Verified Solution • 2025–26 Session moderate

For

x \in \left( {0,{{5\pi } \over 2}} \right),

define

f\left( x \right) = \int\limits_0^x {\sqrt t \sin t\,dt.}

Then

f

has

Accepted Answer

f'\left( x \right) = \sqrt x \sin x

At local maxima or minima,

f'\left( x \right) = 0

\Rightarrow x = 0

or

sin

x=0

\Rightarrow x = 2\pi ,\,\,\pi \in \left( {0,{{5\pi } \over 2}} \right)

f''\left( x \right) = \sqrt x \cos \,x + {1 \over {2\sqrt x }}\sin \,x

= {1 \over {2\sqrt x }}\left( {2x\,\cos \,x + \sin \,x} \right)

At

x = \pi ,

f''\left( x \right) &lt; 0

Hence, local maxima at

x = \pi

At

x = 2\pi ,\,\,\,f''\left( x \right) &gt; 0

Hence local minima at

x = 2\pi

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