Find the constant $a$ so that the max rate of change of $T$ at $P(1,1,1)$ equals 3x the rate at which $T$ is changing in the $y$ direction

Question

Suppose that the temperature $T$ in $R^3$ is given by $T(x,y,z)=x^2+y+az^3$. Find the constant $a$ so that the maximum rate of change of $T$ at $P(1,1,1)$ equals three times the rate at which $T$ is changing in the $y$ direction

Looking for some help with this exam review question.

Answer 1

Hint 1: The maximum rate of change is in the direction of $\nabla T$ and the rate of change is $|\nabla T$|.

Hint 2: The rate of change of temperature in the $y$ direction is given by the directional derivative $D_y T = \hat j \cdot \nabla T$.