Question

How do you find all local maximum and minimum points given `y=x^2-x`?

Answer 1

See explanation.

Explanation:

To find the local extreme (maximum or mininmum) of a function `f(x)` we have to find its derivative and check where it is zero.

Here we have:

`f(x)=x^2-x`

`f'(x)=2x-1`

`2x-1=0 iff x=1/2`

The point `x_0=1/2` can be a critical point of `f(x)`. To check if it is a critical point we have to find out if the derivative changes sign at `x_0=1/2`

graph{2x-1 [-2.738, 2.737, -1.37, 1.367]}

As we can see the derivative changes sign from negative to positive, so the point `x_0=1/2` is a minimum.

Answer:

The function `f(x)=x^2-x` has a minimum at `x_0=1/2`.