## Calculus (3rd Edition)

The curves $y=x^{2}$ and $y=\cos x$ intersect.
Let $f(x)=x^{2}-\cos x .$ Note that any root of $f(x)$ corresponds to a point of intersection between the curves $y=x^{2}$ and $y=\cos x .$ Now, $f(x)$ is continuous over the interval $$\left[0, \frac{\pi}{2}\right], f(0)=-1\lt0$$ and $$f\left(\frac{\pi}{2}\right)=\frac{\pi^{2}}{4}\gt0$$ Therefore, by the Intermediate Value Theorem, there exists a $c \in\left(0, \frac{\pi}{2}\right)$ such that $f(c)=0 ;$ consequently, the curves $y=x^{2}$ and $y=\cos x$ intersect.