Question
Download Solution PDFp का वह मान क्या है जिसके लिए फलन \(\rm f(x)= p \sin x + \dfrac{\sin 3x}{3}\) में \(\rm x=\dfrac{\pi}{3} \) पर चरममान है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFफलन:
यदि फलन f(x) में x = a पर चरममान है, तो f'(a) = 0 है।
गणना:
दिया गया है, फलन \(\rm f(x)= p \sin x + \dfrac{\sin 3x}{3}\) है।
⇒ f'(x) = \(\rm p\;cos \; x + \dfrac {3\;cos \;3x}{3}\)
⇒ f'(x) = \(\rm p\;cos \; x + cos \;3x\)
⇒ f'(\(\rm \dfrac {\pi}{3}\)) = \(\rm p\;cos \; (\dfrac {\pi}{3}) + cos \;3(\dfrac {\pi}{3})\)
⇒ f'(\(\rm \dfrac {\pi}{3}\)) = \(\rm p\;cos \; (\dfrac {\pi}{3}) + cos \; \pi\)
फलन \(\rm f(x)= p \sin x + \dfrac{\sin 3x}{3}\) में \(\rm x=\dfrac{\pi}{3}\) पर चरममान है।
इसलिए, \(\rm f'(\dfrac {\pi}{3}) = 0\)
⇒ \(\rm p\;cos \; (\dfrac {\pi}{3}) + cos \; \pi\) = 0
⇒ \(\rm \dfrac p 2 -1 = 0\)
⇒ \(\rm \dfrac p 2 =1\)
⇒ \(\rm p = 2\)
अतः p का वह मान 2 है, जिसके लिए फलन \(\rm f(x)= p \sin x + \dfrac{\sin 3x}{3}\) में \(\rm x=\dfrac{\pi}{3}\) पर चरममान है।
Last updated on May 19, 2025
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