Answer
See explanations.
Work Step by Step
Step 1. Given the equation $x+2cos(x)=0$. Letting $f(x)=x+2cos(x)$, we can identify it is a continuous function in its domain.
Step 2. Evaluate function values at two different points: $f(0)=0+2cos(0)=2\gt0$ and $f(-\pi/2)=-\pi/2+2cos(-\pi/2)=-\pi/2\lt0$
Step 3. Since the function values of the two points are on the opposite sides of the $x$-axis, using the Intermediate Value Theorem, the function must cross the $x$-axis at least once. In other words, there must be a point $x=c$ in $(-\pi/2,0)$ to give $f(c)=0$ which gives at least one solution to the equation.
