sketch a graph of a function that satisfies all of the given conditions

Question

f′(0) = f′(2) = f′(4) = 0,

f′(t) > 0 if t < 0 or 2< t < 4

f′(t) < 0 if 0 < t < 2 or t > 4

f′′(t) > 0 if 1 < t < 3

f′′(t) < 0 if t < 1 or t > 3.

Answer 1

What have you tried?

f′(0) = f′(2) = f′(4) = 0,

The slope for f(x) at x=0, 2 and 5 is 0. That is the slope is horizontal. That is there is a max or min at these points.

f′(t) > 0 if t < 0 or 2< t < 4

The function f(x) is increasing on those intervals

f′(t) < 0 if 0 < t < 2 or t > 4

The function f(x) is decreasing on those intervals

f′′(t) > 0 if 1 < t < 3

f(x) is concave up on these intervals

f′′(t) < 0 if t < 1 or t > 3

f(x) is concave down on these intervals