What is the value of p for which the function \(\rm f(x)= p \sin x +

1.	What is the value of p for which the function \(\rm f(x)= p \sin x + \dfrac{\sin 3x}{3}\) has an extremum at \(\rm x=\dfrac{\pi}{3} \ ?\)
Answer» Correct Answer - Option 4 : 2 Concept: If the function f(x) has an extremum at x = a then f'(a) = 0 Calculations: Given, the function is \(\rm f(x)= p \sin x + \dfrac{\sin 3x}{3}\) ⇒ 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'(\(\rm \dfrac {\pi}{3}\)) = \(\rm p\;cos \; (\dfrac {\pi}{3}) + cos \; \pi\) The function \(\rm f(x)= p \sin x + \dfrac{\sin 3x}{3}\) has an extremum at \(\rm x=\dfrac{\pi}{3}\) Therefore, \(\rm f'(\dfrac {\pi}{3}) = 0\) ⇒ \(\rm p\;cos \; (\dfrac {\pi}{3}) + cos \; \pi\) = 0 ⇒ \(\rm \dfrac p 2 -1 = 0\) ⇒ \(\rm \dfrac p 2 =1\) ⇒ \(\rm p = 2\) Hence, the value of p for which the function \(\rm f(x)= p \sin x + \dfrac{\sin 3x}{3}\) has an extremum at \(\rm x=\dfrac{\pi}{3}\) is 2.

What is the value of p for which the function \(\rm f(x)= p \sin x + \dfrac{\sin 3x}{3}\) has an extremum at \(\rm x=\dfrac{\pi}{3} \ ?\)

Answer» Correct Answer - Option 4 : 2

Concept:

If the function f(x) has an extremum at x = a then f'(a) = 0

Calculations:

Given, the function is \(\rm f(x)= p \sin x + \dfrac{\sin 3x}{3}\)

⇒ 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'(\(\rm \dfrac {\pi}{3}\)) = \(\rm p\;cos \; (\dfrac {\pi}{3}) + cos \; \pi\)

The function \(\rm f(x)= p \sin x + \dfrac{\sin 3x}{3}\) has an extremum at \(\rm x=\dfrac{\pi}{3}\)

Therefore, \(\rm f'(\dfrac {\pi}{3}) = 0\)

⇒ \(\rm p\;cos \; (\dfrac {\pi}{3}) + cos \; \pi\) = 0

⇒ \(\rm \dfrac p 2 -1 = 0\)

⇒ \(\rm \dfrac p 2 =1\)

⇒ \(\rm p = 2\)

Hence, the value of p for which the function \(\rm f(x)= p \sin x + \dfrac{\sin 3x}{3}\) has an extremum at \(\rm x=\dfrac{\pi}{3}\) is 2.