UPSC NDA Math Model Paper 9

Given: f (x) = sin x + cos x
f′(x)=

d

dx

(f(rm‌x))
=

rm‌d

dx

(sin‌x+cos‌x)
=cos‌x−sin‌x
Now, f′′(x)=

d

dx

(f′(x))
=

d

dx

(cos‌x−sin‌x)
=−sin‌x−cos‌x
=−(sin‌x+cos‌x)
Here, f’’(x) < 0, so given function is maxima
∴ f’(x) = 0
⇒cos‌x−sin‌x=0
⇒cos‌x=sin‌x
We know at x = π/4, sin x = cos x
f (x) = sin x + cos x
= sin x + sin x (∵ sin x = cos x)
= 2 sin x
F(π/4) = 2 sin(π/4)
=2(

1

√2

)
Divide and multiply by √2 , we get
=√2
Hence, option (2) is correct.