Consider the function f(x) = sin(x) in the interval x in [pi/4, 7pi/4]. The…
2012
Consider the function f(x) = sin(x) in the interval x in [pi/4, 7pi/4]. The number and location(s) of the local minima of this function are
- A.
One, at π/2
- B.
One, at 3π/2
- C.
Two, at π/2 and 3π/2
- D.
Two, at π/4 and 3π/2
Show answer & explanation
Correct answer: D
Key idea: find critical points from f'(x)=cos x and check endpoint behavior.
Step 1: Compute derivative. f'(x)=cos x so cos x=0 gives critical points x=π/2 and x=3π/2 in the interval.
Step 2: Classify critical points using f''(x) = -sin x. At x=π/2, f''(π/2) = -1 < 0, so π/2 is a local maximum. At x=3π/2, f''(3π/2) = 1 > 0, so 3π/2 is a local minimum.
Step 3: Check endpoints. At the left endpoint x=π/4, f'(π/4)=cos(π/4)>0, so the function increases to the right of π/4 and π/4 is a one-sided local minimum. The right endpoint x=7π/4 is not a local minimum.
Conclusion: The function has two local minima on [π/4, 7π/4], located at x = π/4 and x = 3π/2.