Finding a free maximum Derivatives and extreme points Using second derivatives to identify a maximum/minimum

Finding a free maximum At this point: We know how to calculate the slope of a function of one or several variables This is done with the derivatives of the functions There are as many derivatives as there are variables in the function Now we know how to calculate slopes, we need to bring all this together to identify a potential maximum or minimum We already know this has something to do with being at a point where the slope is zero.

Finding a free maximum Extreme points of functions of one or several variables Identifying maximum and minimum

Extreme points of functions y x B A s 1 C 1 -s Slope > 0 Slope < 0 Slope = 0 Reminder: For a function of 1 variable, the extreme points occur where the slope is zero

Extreme points of functions This means that in order to find the extreme point of a function of a single variable, we first take the 1st derivative: The maximum of f(x) occurs when f’(x)=0 So we set the derivative equal to zero and solve for x, using the methods we used in the “algebra” section.

Extreme points of functions y x For a function of 2 variables, the extreme points occur where the slope is zero both in the x- direction and the y- direction In other words, both partial derivatives must be equal to zero

Extreme points of functions This means that in order to find the extreme point of a function of several variables, we first take the partial derivatives: We then set these to zero to find the extremum

Extreme points of functions Setting the derivatives equal to zero gives a system of N equations with N unknowns: This can be solved using the methods we saw in the algebra part of the course

Finding a free maximum Extreme points of functions of one or several variables Identifying maximum and minimum

x Minimum Problem: The slope of a function is zero, both for a maximum and a minimum The zero-derivative condition does not help us here !! So how do we find which is which ? y Maximum y x

Identifying maximum and minimum x Minimum Maximum: The function is first increasing, then decreasing Therefore, the slope of the function is decreasing There is a maximum when the second derivative is negative y Maximum y x Minimum: The function is first decreasing, then increasing Therefore, the slope of the function is increasing There is a minimum when the second derivative is positive

Identifying maximum and minimum Using the example we had above: Do we have a maximum or a minimum? Positive, so we have a minimum

Identifying maximum and minimum Things become a bit more complicated for functions of several variables. The general idea is still to look at the sign of the second derivative. Only this time, because there are several partial derivatives, there is not a single ‘second derivative’ This is called the Hessian matrix

Identifying maximum and minimum In general, with a function This matrix is given by: This means “take the 1 st partial derivative with respect to x, and differentiate with respect to y This means “take the 1 st partial derivative with respect to x, and differentiate with respect to x again

Identifying maximum and minimum To understand, let’s use an example: So:

Identifying maximum and minimum In our example, the matrix is given by: Step 1: calculate the ‘cross product’ at the solution point, to check that : If its negative, the solution is neither a maximum, nor a minimum: it is beyond this course!

Identifying maximum and minimum In our example, the matrix is given by: Step 2: Check the sign of the top left entry of the matrix, like the single function case If it is positive, you have a minimum If it is negative, you have a maximum In this case it is positive, so we have a minimum