Let f and g be functions such that f prime (x) and g prime (x) are both continuous and increasing
for all real numbers x. Which of the following is ALWAYS true?

I. f + g is concave up everywhere.
II. f - g is concave up everywhere.
III. f multiplied by g is concave up everywhere.

Question

Let f and g be functions such that f prime (x) and g prime (x) are both continuous and increasing
for all real numbers x. Which of the following is ALWAYS true?


I. f + g is concave up everywhere.
II. f - g is concave up everywhere.
III. f multiplied by g is concave up everywhere.

anonymous · Answer

The first one is definitely always true. The second one is not always true. The third I'm not completely sure on. Negative values of f and g make it a bit messy, but I would be inclined to say that it is not always true.

anonymous · Answer

any 
explanation for it is either true or false would be great

anonymous · Answer

If f'(x) and g'(x) are both everywhere continuous and increasing, then (f'+g')(x) is also everywhere continuous and increasing. If a function's derivative is everywhere continuous and increasing, then that function is concave up everywhere.

The reason that two and three are sometimes false is because (f'-g')(x) and (fg)'(x) are not necessarily everywhere continuous and increasing.