Question

Can a function be concave down and positive everywhere?

Can a function be increasing and be concave down everywhere?

Can a function have two local extrema and three inflection points?

Can a function have 4 zeros and two local extrema?

Please provide examples to help me visualize, thanks!

Answer 1

1) No. For example
\[ f(x) = - x^2 + C \]
No matter what value of C you have, there will be an x that will eventually be negative...

Answer 2

2) No. Again, using f(x) from above, for part of the domain, f is increasing, but eventually, the derivative of f (which is monotonically decreasing) will go negative. The derivative is -x . No matter what -x will eventually go below zero...

Answer 3

3) If you mean ONLY and exactly two local maxima and exactly three inflection points, then you you can always cook up something. So you need some information about continuity and smoothness, but yes you can do this and 4)