Conjecture: If second derivative of a function is positive everywhere then it has no absolute maximum.

IncludedExcluded 06/12/2018. 3 answers, 307 views

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

3 Answers

Ted Shifrin 06/12/2018.

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.

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.

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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