Finding the Local max and min

[Hockeyman]

The questions reads:
Find values of "a" and "b" so that the function f(x)=x^2 + ax + b has a local minimum at the point (6,-5). How would you start this one?

Find the value of "a" so that the function f(x)=xe^ax has a critical point at x=3
Again i'm not really sure how to start this, I know the answer is 1/3 but how did they get that?

 

[galactus]

The questions reads:
Find values of "a" and "b" so that the function f(x)=x^2 + ax + b has a local minimum at the point (6,-5). How would you start this one?

You are given x=6 and y=-5

\(\displaystyle \L\\(6)^{2}+a(6)+b=-5\)

The derivative is 0 at x=6, in order to have a critcal point there.

\(\displaystyle \L\\f'(x)=2x+a\)

\(\displaystyle \L\\2(6)+a=0\)

You have two equations:

\(\displaystyle \L\\6a+b=-41\)
\(\displaystyle \L\\a=-12\)

You can use the same method on the last problem.

EDIT: Sorry, I got the coordinates on the graph backwards.