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

Step 1: Set the two graphs equal.
Intersection points satisfy $f(x)=g(x)$, that is $2\cos\left(\frac{x}{2}\right)+3=4$.
Step 2: Isolate the cosine.
Subtract $3$ and divide by $2$: $\cos\left(\frac{x}{2}\right)=\frac12$.
Step 3: Substitute to simplify.
Let $\theta=\frac{x}{2}$, so we solve $\cos\theta=\frac12$.
Step 4: Translate the interval.
As $x$ runs over $[0,4\pi]$, $\theta=\frac{x}{2}$ runs over $[0,2\pi]$.
Step 5: Solve within $[0,2\pi]$.
$\cos\theta=\frac12$ has solutions $\theta=\frac{\pi}{3}$ and $\theta=\frac{5\pi}{3}$, both inside the interval. That is two values.
Step 6: Convert back and count.
These give $x=\frac{2\pi}{3}$ and $x=\frac{10\pi}{3}$, both in $[0,4\pi]$. So there are exactly $2$ intersection points.
\[ \boxed{2} \]