Q: If $\large f(x) = (sin^2 x- 1)^n (2 + cos^2 x)$ , then x = π/2 is a point of

(A) local maximum, if n is odd

(B) local minimum, if n is odd

(C) local maximum, if n is even

(D) local minimum, if n is even

Sol. If x = a is the point of local extremum of

y = f(x), then f( a –h).f(a +h) > 0

⇒ f( π/2 –h) . f( π/2+h) > 0

f( π/2- h) = (- ve)^{n} ….(1)

f( π/2 +h) =( -ve)^{n} ….(2)

f(π/2) = 0 ….(3)

⇒ f( π/2 – h).f(π/2 +h) = ( -ve )^{2n} > 0

⇒ n can be odd or even.

So from (1),(2) and (3),if n is odd or even maxima or mimima occurs accordingly .

Hence (A), (D) are correct.