Answer
The equation has 1 solution: $(0.739)$.
Work Step by Step
$$\cos x=x$$ $$\cos x-x=0$$
To prove the equation has a solution using the Immediate Value Theorem, we need to find 2 values $x_1$ and $x_2$ and show that there is a change of sign in the function $f(x)=\cos x-x$ as $x$ goes from $x_1$ to $x_2$.
- Take $x_1=0$: $f(0)=\cos0-0=1-0=1\gt0$
- Take $x_2=\pi/2$: $f(\pi/2)=(\cos\pi/2)-\pi/2=0-\pi/2=-\pi/2\lt0$
So there is a change of sign in $f(x)$ as $x$ goes from $0$ to $\pi/2$.
- On $[0,\pi/2]$: $\lim_{x\to c}f(x)=\lim_{x\to c}(\cos x-x)=\cos c-c=f(c)$
So $f(x)$ is continuous on $[0,\pi/2]$.
Therefore, according to the Intermediate Value Theorem, there must be a value $x=c\in[0,\pi/2]$ such that $f(c)=0$. In other words, the equation $\cos x-x=0$ or $\cos x=x$ has at least one solution in the interval $[0,2]$.
The graph of the function $f(x)=\cos x-x$ is enclosed below. Looking at the graph, it turns out that the curve $f(x)$ crosses the line $y=0$ at $1$ point. The $x$-coordinate of this point is the solution of the equation.
So the equation has 1 solution: $(0.739)$
