- #1
Contingency
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Homework Statement
Givens: [tex]\forall x\ge 0:\quad f^{ \prime \prime }\left( x \right) \ge 0;\quad f\left( 0 \right) =0[/tex]
Prove: [tex]\forall a,b\ge 0:\quad f\left( a+b \right) \ge f\left( a \right) +f\left( b \right)[/tex]
Homework Equations
By definition, f is convex iff [tex]\forall x,y\in \Re \quad \wedge \quad \forall \lambda :\quad 0\le \lambda \le 1\quad \Rightarrow \quad f\left( \lambda x+(1-\lambda )y \right) \le \lambda f\left( x \right) +(1-\lambda )f\left( y \right)[/tex]
The Attempt at a Solution
Intuition-wise I see that a convex function's values increase at an increasing rate, but that's equivalent to [tex]f^{ \prime \prime }\left( x \right) \ge 0[/tex]
I also see that [tex]f\left( 0 \right) =0[/tex] is necessary for the inequality to hold, but I can't find any tools with which I can work on proving the inequality.
Also I figure [tex]\forall x\ge 0:\quad f^{ \prime }\left( x \right) \ge 0[/tex] and also monotonously increasing.
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