$(10$ pts $)$ Consider the function

$f(x,y,z)=x^2{e}^{y{z}^2}$

at the point $P(2,0,4).$

$($a$)(3$ pts$)$ What is the maximal rate of change of $f$ at $P?$

$($b$)(3$ pts $)$ In what direction is the maximal rate of change?

$($c$)(4$ pts $)$ Determine the general equation of the tangent plane at the point $P.$

$5$