I am studying calculus but came across a problem which I believe needs the above theorem. I am at a loss....
HINT: If the function $f$ is everywhere twice-differentiable, at a local (or absolute) maximum point $x=a$, you must have $f''(a)\le 0$.
I assume that you're asking about a function defined (and twice differentiable) on all of $\Bbb R$.
Hint: By the second derivative test, any critical point will be a local minimum
Differentiating a function tells you its gradient function, inputting a value into this function gives you the gradient at some point.
Differentiating the gradient function tells you what is happening the gradient, that is: whether it is increasing or decreasing, and at what rate. If the second derivative is positive everywhere, it implies that the gradient never decreases, and therefore that the curve never slopes down - which means it has no maximum.