Can somebody please help me with this derivatives problem involving graphs?

Question

Suppose f and g are both concave upward on (-infinity, infinity). Under what condition on f will the composite function
h(x)= f(g(x)) be concave upward?

Answer 1

Well we know that f'' and g'' are positive. Let's figure out when h'' is too.

h' = f'(g)*g'

h'' = f''(g)*g'*g' + f'(g)*g''

(The Chain Rule and Product Rule are our friends.)

The first of the two terms is always positive. The second will be positive if f' is.

So if f is increasing, we're in great shape. That's not a necessary condition, but it sure is sufficient.

Answer 2

i'm having hardship with the "His velocity is proportional to the area he nevertheless has to bypass" part of the question. i could think of that it potential great velocity on the commencing up then decelerating to 0 on the end. almost it particularly is impossible, commencing at infinite velocity! Are you soliciting for the form of a graph for the above state of affairs - if so then x would be asymptotic to the y axis at time 0. and apart from the x axis at time end (or close to adequate) no longer almost possible. My feeleing right here could be simpler: A graph showing acceleration to the mid distance factor the decelerating to the end. consequently the time taken to the mis factor could be [t(end) - t(initiate)}/2 and x = y = 4 hundred nevertheless puzzled on information this question