Question

Consider graph of function \(f(x)=2\cos\left(\frac{x}{2}\right)+3\), \(g(x)=4\). The number of points of intersection of two graphs in interval \([0,4\pi]\) is:

• A. \(1\)
• B. \(3\)
• C. \(2\)
• D. \(4\)

Hint:

Convert the intersection problem into solving a trigonometric equation and then count all valid solutions in the given interval.

Answer 1

The Correct Option is C

Concept: Points of intersection satisfy \[ f(x)=g(x). \] Therefore, \[ 2\cos\left(\frac{x}{2}\right)+3=4. \]

Step 1: Solve the trigonometric equation.
\[ 2\cos\left(\frac{x}{2}\right)=1 \] \[ \cos\left(\frac{x}{2}\right)=\frac12. \] Let \[ \theta=\frac{x}{2}. \] Then \[ \cos\theta=\frac12. \] General solutions are \[ \theta=2n\pi\pm\frac{\pi}{3}. \]

Step 2: Apply interval restriction.
Since \[ 0\le x\le4\pi, \] we get \[ 0\le \theta\le2\pi. \] Within \([0,2\pi]\), \[ \theta=\frac{\pi}{3},\quad \frac{5\pi}{3}. \] Thus \[ x=\frac{2\pi}{3},\quad \frac{10\pi}{3}. \] Hence there are exactly \[ \boxed{2} \] points of intersection.