Question

How do you find the local extrema for `f(x)=5x-x^2`?

Answer 1

I found `x=5/2 and y=25/4` as coordinates of a point of maximum for our function.

Explanation:

We can derive our function (representing a Parabola) and set the derivative equal to zero: this will give us the `x` coordinate where the function neither increases nor decreases.
So:
`f'(x)=5-2x`
set equal to zero:
`5-2x=0`
and:
`x=5/2`
which corresponds to a `y` value of:
`f(5/2)=5(5/2)-(5/2)^2=25/2-25/4=25/4`

Graphically:
graph{5x-x^2 [-20.28, 20.28, -10.14, 10.14]}