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+13f(x)=37x27x+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.