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Unformatted text preview: strictly concave. If the function is convex then If then the function is convex. If then the function is strictly convex. If then the function is both concave and convex. If the function is both concave and convex then . (c) What’s a uni-variate function which is both concave and convex?
A linear function like
and convexity for all values of . () . For linear functions, (d) True or false: if the function ( ) is concave then the function so it satisfies the definition of concavity ( ) is convex (and vice versa)? True. If ( ) is concave then ( )
for all values of . Now, the second derivative of
which is the definition of a convex function.
() (e) Suppose
() ( ) For example, () ( ) is also concave? If ( ) is concave then
that: () ⏟ () () However, this doesn’t mean that () () ⏟( )
Notice that if ( ) is also concave. To see this, notice () ()
() () ⏟ Suppose the function ( ) is concave. Does this mean that () We know that ( ) is () so that: then – () which means that:
⏟ Thus, if ( ) is concave and ( ) is convex then () ⏟( ) ( ) will be concave. Here are some examples: ⏟
() ⏟ ⏟...
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This homework help was uploaded on 02/15/2014 for the course E 204 taught by Professor Ajaz during the Winter '13 term at University of Toronto- Toronto.
- Winter '13