Math Proofs

[jiggamanjatt7]

1. suppose the f(x) has a continuous first derivative for all xER.

a) prove that f(x) is concave if and only if f(x*) +(x-x*)f'(x*)is less than or equal to f(x) for all x and x*ER.

b)given that f(x) is concave, prove the x* is a global maximum of f(x) if and only if f'(x*)= 0.

c) given that f(x) is concave, prove that its set of global maxima is either empty, a singleton, or an infinite convex set


2. a) given f: R -> R, prove that differentiability implies continuity but not vice versa.

b)consider an interval I c R and suppose that f: I -> R is continuous. prove that if i) x* is a local maximum of f and ii) x* is the only extreme point of f on I, then x* is a global maximum of f on I.

 

[pka]

What have you tried?
Where do you have trouble?

 

[galactus]

For #2: Try using the defintion of continuity. A function is continuous if

1. f(c) is defined

2. \(\displaystyle \lim_{h\to{0}}f(c+h)\) exists

3. \(\displaystyle \lim_{h\to{0}}f(c+h)=f(c)\)

Now, show that \(\displaystyle \lim_{h\to{0}}f(x_{0}+h)=f(x_{0})\)

or equivalently

\(\displaystyle \lim_{h\to{0}}\left[f(x_{0}+h)-f(x_{0})\right]=0\)

Just a few steps.

 

[jiggamanjatt7]

This is all i have done so far


FOR QUESTION 1

x* is simply a name for a real number, fixed for the moment. In the
statement

f(x*) +(x-x*)f'(x*)<= f(x)

x* is arbitrarily chosen and fixed while x varies. The statement says that
the entire graph of f lies above every tangent line. If you are to get
started on this, you need first to write down the definition of 'f is
concave'. And then you need to relate this definition to the statement you
are trying to prove.


FOR QUESTION 2

a) Look at the definitions of 'differentiable at a point' and 'continuous
at a point'. One of them has more requirements than the other. That is to
say, one of the definitions is 'contained' in the other. Use this to do the
'prove'. For the 'not vice versa', come up with a function which you can
show to be continuous at some specific point, but which you can also show
not to be differentiable at that point.

b) What are the definitions of 'local maximum', 'extreme point on an
interval', and 'global maximum'?

 

[stapel]

This is all i have done so far...

All you have shown in your most-recent post is that you've scraped the reply you were provided elsewhere and claimed it to be your own work. :?

What have you done?

Eliz.