find the constant a,b and c so that the direction derivative of Φ=axy^2 +byz+cz^2 x^3 at (1,2,−1) has a maximum of magnitude 64 jn a direction parallel to the z axis.


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Question Number 65797 by Souvik Ghosh last updated on 04/Aug/19

$f i n d t h e c o n s t a n t a, b a n d c s o$ $t h a t t h e d i r e c t i o n d e r i v a t i v e o f$ $Φ = {a x y}^{2} + b y z + {c z}^{2} x^{3} a t (1, 2, - 1)$ $h a s a m a x i m u m o f m a g n i t u d e$ $64 j n a d i r e c t i o n p a r a l l e l t o t h e$ $z a x i s .$
	Answered by Tanmay chaudhury last updated on 04/Aug/19

	$▽ Φ$ $(i \frac{\partial}{\partial x} + j \frac{\partial}{\partial y} + k \frac{\partial}{\partial z}) ({a x y}^{2} + b y z + {c z}^{2} x^{3})$ $i ({a y}^{2} + 3 x^{2} {c z}^{2}) + j (2 a x y + b z) + k (b y + 2 {c x}^{3} z)$
	Commented by Tanmay chaudhury last updated on 04/Aug/19

	$e a i t i h a v e c o r r e c t e d m y e r r o r d e t e c t e d b y y o u . . t h a n k y o u$ $w a i t i a m t r y i n g$
	Commented by Souvik Ghosh last updated on 04/Aug/19

	$b u t s i r h e a r$ $\vec{▽} Φ = ({a y}^{2} + 3 {c z}^{2} x^{2}) i + (2 a x y + b z) j$ $+ (b y + 2 {c z x}^{3}) k$
	Commented by Tanmay chaudhury last updated on 04/Aug/19

	$y e s y o u a r e r i g h t . . . i n h u r r y$
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