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

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Answer 1 · 2015-02-25T15:22:46

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 $y$

$y = 2$

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