Question

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

Answer 1

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.