I am studying calculus but came across a problem which I believe needs the above theorem. I am at a loss....

Ted Shifrin 06/12/2018 at 18:49.

**HINT**: If the function $f$ is everywhere twice-differentiable, at a local (or absolute) maximum point $x=a$, you must have $f''(a)\le 0$.

Omnomnomnom 06/12/2018 at 18:49.

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

Rhys Hughes 06/12/2018 at 18:59.

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.

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