Functions Where Every Local Minimum Is Global

Detailed Solution

Take $f(x)=\sin x$. It is continuous, and its local minima occur at $x=-\pi/2+2k\pi$, $k\in\mathbb Z$, each with value $-1$, which is exactly the global minimum of $f$ over $\mathbb R$. So every local minimum is indeed a global minimum, yet $f$ has infinitely many local minima, all sharing the same value $-1$ — this confirms (C) and simultaneously rules out (A), which wrongly insists on exactly one local minimum. It also rules out (B): $\sin x$ oscillates rather than decreasing once and then increasing once. Finally, $\sin x$ is certainly not monotone on $\mathbb R$, ruling out (D). Thus the correct option is $\boxed{\mathrm{C}}$.