Partial Derivitives

[Administrator]

When working with critical angles of partial derivatives, if they don’t show up in the simultaneous equation of f'(x)=f'(y)=0 then are no more critical values? Because I’ve found the critical values in an equation and was wondering if those I have with that method would be all of them?
And so are critical values are only found through the simultaneous equation when dealing with partial derivatives?

[Anonymous]

Hi Ben,
For determining critical values, you follow the process in the image attached. Essentially, you first partially differentiate with respect to x and y and solve for when these are both equal to zero. This will usually give simultaneous equations as a result, which you next solve to find a number of (x,y) points, which are defined as the critical points of the function. Each critical point corresponds to a unique critical value which is the result of plugging the points into the original equation. So, if you have found there is only two critical points, then there must be only two critical values for this function. These critical values will either represent local minimums, local maximums or inflection points when graphed in 3D, depending on the original function. Keep in mind some questions may ask you to find global maximum and minimum points for the function over a defined range, which could be larger or smaller than the critical values. This means you are also required to check the values of the function at all of the boundary points for the function, and determine the overall smallest or largest value.

[Author]

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