How do you find the coordinates of the local extrema of the function?

1 Answer
Sep 7, 2014

The best way to do this is to find the derivative of the function.

Let's say our function is #f(x)=3/7x^2-7x+13#. Not a particularly pretty or elegant function, but we can still find its minimum.

First, we take the derivative of the function. Remember:
#"if "f(x)=x^a," then "f'(x)=ax^(a-1)#
So: #f'(x) = 6/7x-7#

Now we need to find the point where #f'(x) = 0#, i.e. the slope of the original function is 0.

So #6/7x-7 = 0#
#=>6/7x = 7#
#=> 6x = 49#
#=>x=49/6#

The last step is to plug this x value into the original equation.

#f(49/6)=3/7(49/6)^2-7(49/6)+13#
#=3/7*2401/36-2401/6+13#
#=343/12-343/6+13#
#=343/12-686/12+156/12#
#=-187/12#

So the local minimum is at #(49/6,-187/12)#.

Note that I deliberately didn't pick a "nice and easy" function - this is to show that finding the derivative, and making the derivative equal zero, works for everything.