Consider such that is periodic with period

MATHEMATICS · Single correct · AP · AP Precalculus · Trigonometric and Polar Functions · Trigonometric and Polar Functions

Problem

Consider f(x)={cos⁡x,0≤x<π2(π2−x)2,π2≤x<πf(x) = \begin{cases} \cos x, & 0 \le x < \frac{\pi}{2} \\ \left(\frac{\pi}{2} - x\right)^2, & \frac{\pi}{2} \le x < \pi \end{cases} such that ff is periodic with period π\pi , then (1) the range of ff is [0,π24]\left[0, \frac{\pi^2}{4}\right] (2) ff is continuous for all real xx , but not differentiable for some real xx (3) ff is continuous for all real xx (4) the area bounded by y=f(x)y = f(x) and the XX -axis from x=−nπx = -n\pi to x=nπx = n\pi is 2n(1+π224)2n \left(1 + \frac{\pi^2}{24}\right) for a given n∈Nn \in N

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