UPSC NDA Math Model Paper 10

Consider, the function f(x)=p‌cos‌x−

cos‌3‌x

3

Taking derivative w.r.to x, we get
⇒f′(x)=−p‌sin‌x+

3‌sin‌3‌x

3

⇒f′(x)=−p‌sin‌x+sin‌3‌x
⇒f′(

π

6

)=−p‌sin‌

π

6

+sin‌3‌

π

6

⇒f′(

π

6

)=−

p

2

+1
The function f(x)=p‌cos‌x−

cos‌3‌x

3

has an extremum at x=

π

6

⇒f′(

π

6

)=0
⇒−

p

2

+1=0
⇒p=2
Hence, the value of p for which the function f(x)=p‌cos‌x−

cos‌3‌x

3

has an extremum at x =

π

6

is 2.