Question 11: Suppose the temperature (T) at a point (x, y, z) in \mat…

Suppose the temperature ($T$) at a point $(x, y, z)$ in $\mathbb{R}^3$ is given by the function

$T(x, y, z) = e^{x+y+z}$

Let $T_1$, $T_2$, and $T_3$ be the tangent planes to the function $f(x, y) = \sqrt{14 - x^2 - y^2}$ at the points $(2, 3)$, $(3, 2)$, and $(3, 1)$, respectively. The set of points on each plane $T_1$, $T_2$, and $T_3$ form affine subspaces $A_1$, $A_2$, and $A_3$, respectively, of the vector space $\mathbb{R}^3$, with respect to usual addition and scalar multiplication. Let $V_1$, $V_2$, and $V_3$ denote the vector subspaces corresponding to the affine subspaces $A_1$, $A_2$, and $A_3$, respectively.

Answer the subquestions using the given information.

Which of the following denotes the maximum rate of change of temperature (T) at the point of intersection of the tangent planes T1, T2, and T3?