Maximum speed of a particle given velocity function in terms of vectors Find the maximum speed of a particle whose velocity, $\mathbf v$ m/s at time t seconds is given by: $$v=2\mathbf isin(t)+\mathbf jcos(t)+3\mathbf k, t\ge0$$
How do I solve this? I tried differentiating and equating to zero but I don't know if it's a valid approach here and if it is, how to take it from there.

Hint: Speed is the magnitude of the velocity. So if the velocity is
$$v(t) =(2\sin t,\cos t, 3)$$ then the speed is
$$f(t)=|v(t)|=\sqrt{4\sin^2 t +\cos^2 t + 9}=\sqrt{10+3\sin^2 t}$$
using the facts that $|(a,b,c)|^2=a^2+b^2+c^2$ and $\sin^2 t+\cos^2 t=1$.
Now you have a real-valued function to maximize on $[0,\infty)$, which is a Calculus I problem.
