The problem only gives me information about function f...!!?

Question

The problem only gives the information about f.
(i) f(0)=2
(ii) f'(x)>0 for all x
(iii) The graph of f is concave up for all x>0 and concave down for all x<0
-Let f be the function defined by g(x) = f(x^2)

1. Find g(0)
- I am stuck from setting the eq to be g(0) = f(0^2)

2. Find the x coordinate of all minimum points of g.
- minimum ts.. exist when g' = zero or und.. and changes from negative to positive, but i don't know what g is..!!
3. Where is the graph of g concave up?
- That's when g' is increasing and g''>0, but I have no idea how to derive function g out of information of f.

Answer 1

1. g(0) = f(0^2) = f(0) = 2.

For 2 and 3, you know g = f(x^2). So use the Chain Rule.

g'(x) = f'(x) * 2x.

f'(x) is NEVER ZERO, by (ii). Thus, g'(x) = 0 when and only when x = 0.

g'' (x) = (f'(x)*2x)' =, by the Product Rule, f''(x)*2x + 2f'(x).

f''(x) is positive when x is positive and negative when x is negative. The same is true of 2x. Thus, when x is not 0, f"(x)*2x is either the product of two positives or the product of two negatives, and hence is positive EITHER WAY. And at the only point we didn't discuss, it's 0. Meanwhile, 2f'(x) is always positive. Just g"(x) is always positive as well.

And there you are.