Practice problem
Let f:[0,2π)f:[0, 2\pi)f:[0,2π) via f(x)=3sin(5x)−4cos(3x)f(x) = 3 \sin(5x) - 4 \cos(3x)f(x)=3sin(5x)−4cos(3x). Does there exist y∈[0,2π)y \in [0, 2\pi)y∈[0,2π) such that ∀x∈[0,2π)\forall x \in [0, 2\pi)∀x∈[0,2π), f(y)≥f(x)f(y) \ge f(x)f(y)≥f(x)?
Yes, because EVT tells us so.
Yes, because we can create a continuous function ggg exactly like fff but with domain [0,2π][0, 2\pi][0,2π], then use EVT. And because g(2π)=g(0)g(2\pi) = g(0)g(2π)=g(0), the maximum must be in the interval.
Yes, because we can restrict fff to the domain [0,2π−ε][0, 2\pi - \varepsilon][0,2π−ε] for any ε>0\varepsilon > 0ε>0, then use EVT.
No, because [0,2π)[0, 2\pi)[0,2π) isn't a closed interval.
No, because fff is not continuous.
Type a concept, like “slope” or “photosynthesis”.