Question

How do you find the point on the graph of `x^2−2x=15−y^4` has the largest y-coordinate?

Answer 1

Start by solving a simpler equation:
`x^2−2x = 15−f(x)`
which simplifies to
`f(x) = -x^2 + 2x + 15`

`f(x)` takes on a maximum value when
`f'(x) = 0`
that is when
`-2x + 2 = 0`
`rarr x = 1`

Note that if `f(x) = y^4`
a maximum value of `f(x)` implies a maximum value of `y^4` provided `f(x) >= 1`

Using `x=2` in the original equation gives
`y^4 = 16`
and therefore (since we are only interested in the largest value for `y`

`y = 2`