# Help with some proofs

• May 18th 2009, 12:21 PM
Rylude
Help with some proofs
Suppose that f(x) has a continuous first derivative for all x http://upload.wikimedia.org/math/7/b...e1a4c7cff0.png R

a) Prove that f(x) is concave if and only if f(x*)+(x-x*)f′(x*) ≥ f(x) for all x and x* http://upload.wikimedia.org/math/7/b...e1a4c7cff0.png R

b)Given that f(x) is concave, prove that x* is a global maximum of f(x) is and only if f′(x*)= 0

c) Given that f(x) is strictly concave, prove that it cannot possess more than one global maximum.

my understanding of graphs is very poor any help on this would be greatly appreciated.

Dave
• May 18th 2009, 12:39 PM
HallsofIvy
Quote:

Originally Posted by Rylude
Suppose that f(x) has a continuous first derivative for all x http://upload.wikimedia.org/math/7/b...e1a4c7cff0.png R

a) Prove that f(x) is concave if and only if f(x*)+(x-x*)f′(x*) ≥ f(x) for all x and x* http://upload.wikimedia.org/math/7/b...e1a4c7cff0.png R

Find the equation of the line from (x, f(x)) to (x*,f(x*)) and use the definition of concave.

Quote:

b)Given that f(x) is concave, prove that x* is a global maximum of f(x) is and only if f′(x*)= 0
If f'(x)= 0, there are three possibilities: x* is a local maximum, a local minimum, or a point of inflection. Show that it must be a local maximum and then a global maximum.

Quote:

c) Given that f(x) is strictly concave, prove that it cannot possess more than one global maximum.
Suppose it had two global maximum and show that leads to a contradiction. You might want to look at the line between the two maxima.

my understanding of graphs is very poor any help on this would be greatly appreciated.

Dave[/QUOTE]