86. \( x \)-intercept(s) for the function \( f(x)=\cos 2 x+1 \) is (are): (where \( k \in Z \) ) \( \begin{array}{ll}\text { (A) } x=\pi+2 k \pi & \text { (B) } x=\frac{\pi}{2}+\pi \\ \text { (C) } x=\pi+k \pi & \text { (D) } x=\frac{\pi}{2}+\frac{k \pi}{2}\end{array} \)

To find the \( x \)-intercept(s) of the function \( f(x) = \cos(2x) + 1 \), we need to set the function equal to zero: \[ \cos(2x) + 1 = 0. \] This simplifies to: \[ \cos(2x) = -1. \] The cosine function is equal to -1 at odd multiples of \( \pi \). Therefore, we can express this condition as: \[ 2x = (2k + 1)\pi, \quad k \in \mathbb{Z}. \] Now, solving for \( x \): \[ x = \frac{(2k + 1)\pi}{2} = \frac{\pi}{2} + k\pi. \] Thus, the solution represents an infinite family of \( x \)-intercepts at intervals of \( \pi \) from \( \frac{\pi}{2} \). This corresponds to option \( D \): \[ \text{(D) } x=\frac{\pi}{2}+\frac{k \pi}{2}. \] **First additional information:** Understanding the properties of the cosine function is vital. It oscillates between -1 and 1, with the key points of \( \cos(2x) = -1 \) happening every \( \pi \) radians along the x-axis, emphasizing its periodic nature. This characteristic means we can predict outcomes over an infinite domain! **Second additional information:** If you’re applying the cosine function in real-world scenarios, such as engineering or physics, recognize how phase shifts can affect waveforms. The \( \cos \) function beautifully models everything from sound waves to electrical currents. The intercepts correspond to definitive moments where the wave represents zero displacement. Isn't it fascinating how mathematics resonates in fields beyond its realm?