Question

3. Determine whether the following properties can be satisfied by a function that is continuous on the interval (−∞, ∞). If such a function is possible provide an example or a sketch of the function. If such a function is not possible explain why.

a. A function f is concave down and positive everywhere.

b. A function f is increasing and concave down everywhere.

c. A function f has exactly 2 local extreme and 3 inflection points.

d. A function f has exactly 4 zeros (x-intercepts) and 2 local extreme.

Answer 1

A) A function f is concave down and positive everywhere.

example of a)

can be a semicircle
example, y=4+

B) A function f is increasing and concave down everywhere.

If f is increasing then f must be positive.

Example of b)

exponential decay.

C) A function f has exactly 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

If there are three inflection points, then f" must change signs three times.

D) A function f has exactly four zeros and two local extrema.

Since there are 4 zeros, there are three distinct intervals between zeros, and, thus, there
must be at least 3 local extrema.

example for D)