help me with this derivative question?

Question

Suppose that f'' and g'' exist and that f and g are concave up for all x. Are the statements below true or false. If true explain if false give an counter example.

1. f(x) + g(x) is concave up for all x
2. f(x) - g(x) cannot be concave up for all x

Answer 1

f(x) + g(x) is concave up for all x. Here's an example:

f(x) = e^x and g(x) = e^x

Although they're the same, they're both still function that are concave up for all x. Then when you add them together:

f(x) + g(x) = e^x + e^x = 2(e^x)

When you graph 2(e^x), you see that it is concave up for all x.

Here's another example:

f(x) = x^2 and g(x) = e^x (both are concave up for all x)

When you graph x^2 + e^x, you see that it's concave up for all x.

Even a function f(x) = (x^2) + (e^x) + (4^x), which consists of three functions that are concave up for all x, is concave up for all x.

Answer 2

1) If a function is concave up, its second derivative is positive, so consider f(x) + g(x). The second derivative of this is f''(x) + g''(x), but both of those are positive, so the sum is positive as well.

2) This is false by the same argument. f(x) - g(x) could be concave up or down since the second derivative of g could be greater or less than that of f. If f = g then the difference is zero which is neither concave up or down.

Example: Let f''(x) = 1 and g''(x) = 2. Both are concave up.

f(x) = x^2 + ax + b and g(x) = 2x^2 + dx + c
f-g = -x^2 + (a-d)x + b-c

If you reverse f and g, the difference is concave up.