15.6 Consider the function \( F\left(x_{1}, x_{2}, y\right)=x_{1}^{2}-x_{2}^{2}+y^{3} \).
a) If \( x_{1}=6 \) and \( x_{2}=3 \), find a \( y \) which satisfies \( F\left(x_{1}, x_{2}, y\right)=0 \).
b) Does this equation define \( y \) as an implicit function of \( x_{1} \) and \( x_{2} \) near \( x_{1}=6 \), \( x_{2}=3 \) ?
c) If so, compute \( \left(\partial y / \partial x_{1}\right)(6,3) \) and \( \left(\partial y / \partial x_{2}\right)(6,3) \).
d) If \( x_{1} \) increases to 6.2 and \( x_{2} \) decreases to 2.9 , estimate the corresponding change in \( y \).
