How do I go about this proof? Should I use the definition of convexity:
f(ax+(1-a)y)<=af(x)+(1-a)f(y)?
You know that for 0<a<1, f(ax+(1-a)y)<af(x)+(1-a)f(y).
Since f>0, you can square this inequality to obtain
f²(ax+(1-a)y)<a²f²(x)+(1-a)²f²(y)+2a(1-a)f(x)f(y).
Now remember that 2f(x)f(y)<f²(x)+f²(y) (because (f(x)-f(y))²>0 )
The inequality becomes
f²(ax+(1-a)y)<a²f²(x)+(1-a)²f²(y)+a(1-a)(f²(x)+f²(y))=f²(x)(a²+a-a²)+f²(y)((1-a)²+a(1-a))=
af²(x)+(1-a)f²(y)
and f² is convex by definition.
Note that if f is C², f convex <=> f''>0.
In this case you have a simpler proof
(f²)'=2ff': (f²)''=2f'2+2ff''>0.
(Note that I used < for
$\displaystyle
\leq
$
all over the place)