Give an example of a function that has a local maximumat (3,-3) and one which has a local minimum at (3.-3).

As you can only ask one question at a time your question
has been edited.

A function f(x) has an extreme point when
f'(x) = 0. To find the x-coordinate, the equation f'(x) = 0 has to be solved. If f''(x)
for the value of x obtained is positive, the point has a minimum value, else if it is
negative the point has a maximum value.

• Let the
  function which has a local minimum at (3, -3) be
  f(x).

f''(3) has to be positive. Let it be
2.

f'(x) = 0 should give a solution of x = 3, an example of
this is f'(x) = 2x - 6.

A function which has a value of -3
at x = 3 is x^2 - 6x + 6

The function f(x) = x^2 - 6x + 6
has a local minimum at (3, -3).

• Let the function
  which has a local maximum at (3, -3) be
  f(x).

f''(3) has to be negative. Let it be
-2.

f'(x) = 0 should give a solution of x = 3, an example
of this is f'(x) = -2x + 6.

A function which has a value of
-3 at x = 3 is -x^2 + 6x -12

The function f(x) = -x^2 + 6x
-12 has a local maximum at (3, -3).

A
function with a local maximum at (3, -3) is f(x) = -x^2 + 6x -12 and a function with a
local minimum at (3, -3) is f(x) = x^2 - 6x + 6