Project ForgeConsider the function f(x) = sin(x) in the interval x ∈ [π/4, 7π/4]. The number and location(s) of the local minima of this function are:
- AOne, at π/2
- BOne, at 3π/2
- CTwo, at π/2 and 3π/2
- Two, at π/4 and 3π/2
Explanation
f'(x)=cos(x)=0 at the interior points x=π/2 and x=3π/2. f''(π/2)=−sin(π/2)=−1<0, so π/2 is a local MAXIMUM. f''(3π/2)=−sin(3π/2)=1>0, so 3π/2 is an interior local minimum. But the domain is a CLOSED interval starting at π/4, and f is increasing immediately to the right of π/4 (f'(π/4)=cos(π/4)>0) — since there's no domain to the left of π/4 to compare against, the boundary point π/4 is also a local minimum relative to the restricted domain. So there are two local minima: at π/4 and 3π/2, matching the file's own answer key.
Source: GATE CS 2012 (2012)
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