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

Can a function be concave down and positive everywhere?
can be a semicircle
example, y=4+
`√(4-x^2)`
attachment 1

Can a function be increasing and be concave down everywhere?
no, concave down means increase slope then decrease slope

Can a function have two local extrema and three inflection points?
inflection points are where the concavity changes
it can be at the ends, the middle and the other end
like in atachment 2, the circles are inflection points

Can a function have 4 zeros and two local extrema?
no, as you can see in attachment 3, there can be 3 zeroes at most for 2 local extrema