Question

How do you find the exact relative maximum and minimum of the polynomial function of `y=x^2+1`?

Answer 1

You can do this in two ways: using your imagination, or the mathematical way.

Explanation:

The first way is: `x^2` can never be negative, but it can be `=0`
In this case `y=0^2+1=1`
For every other value `y>1` so `(x=0,y=1)` is a minimum, and not a relative, but absolute minimum.
There is no maximum, because whatever value `x` assumes, `y` will get greater and greater.

The mathematical way is to define the derivative of the function and set it to zero:
`y'=2x=0->x=0`
And then you have to define whether this is a max or min situation by taking the second derivative `y''=2`.
graph{x^2+1 [-12.61, 12.7, -2.45, 10.21]}