When I use Desmos to draw the graph of the function $f$ defined by $f(x):=x^2-\cos x$, the graph looks very similar to a quadratic function. Unlike the graph of, say, $x-\cos x$, it does not have any wiggles/waves. I would expect the $\cos x$ term in $f(x)$ to cause there to be some wiggles, but this appears not to be the case. Why is this? I assume it has something to do with $x^2$ growing much faster than $\cos x$ ever does. Or perhaps it's because $f$ is everywhere convex, so always keeps growing steeply upwards when $x$ is positive.
I have also notices that the graphs of $x^2-\cos(kx)$ are wiggly for $k\ge3$, but not for $k=1,2$.
Can anyone shed any more light on why $x^2-\cos x$ doesn't have any wiggles?
Note that the second derivative is always positive: $$f(x)=x^2-\cos(x)\implies f''(x)=2+\cos(x)\geq 1>0$$ and therefore the function $f$ is strictly convex (and it doesn't have any wiggles). More generally, $$f_k(x)=x^2-\cos(kx)\implies f_k''(x)=2+k^2\cos(kx)$$ and $f_k''$ changes its sign for $k^2>2$.