ANSWER
$59
EXPLANATION
John's profit is represented by a quadratic function, whose leading coefficient is negative. This means that the graph is a parabola that opens downward and, therefore, the vertex is a maximum. In other words, the most profit he can earn is the y-coordinate of the vertex.
The x-coordinate of the vertex of a quadratic function given in standard form is,
[tex]\begin{gathered} f(x)=ax^2+bx+c \\ x_{vertex}=\frac{-b}{2a} \end{gathered}[/tex]
In this case, a = -1 and b = 10,
[tex]x_{vertex}=\frac{-10}{2(-1)}=\frac{-10}{-2}=5[/tex]
And the maximum profit is given by P(x_vertex),
[tex]P(x_{vertex})=P(5)=-5^2+10\cdot5+34=-25+50+34=59[/tex]
Hence, the most he can earn is $59.
