How Does Quadratic Approximation Determine Local Max/Min?

[Chadlee88]

hey can som1 please help, i know how to find the quadratic approximation for a given function but i don't know how the quadratic approximation determines a local max/min This is with regard to multivariable functions. thanks

 

[berkeman]

If you have a quadratic function of one variable, how do you find any maxima and minima of that function? How do you determine if any points you find are maxima or minima? How are derivatives involved?

Now for multivariable functions, you do much the same thing. But what special kind of derivative do you need to use when dealing with multivariable equations? Do you need to do anything special when finding local maxima and minima of multivariable equations?

Hint -- when thinking about multivariable functions, I like to think of a 3-D plot of a function z = f(x,y). Picture a 3-D surface running through the x-y-z cube. Then think about when there are bumps and such in that surface -- how do you find where they are?

 

[HallsofIvy]

It might help to recall that a graph of a quadratic function is a parabola.

Do you remember completing the square to find the vertex of a parabola?