# find-the-constant-a-b-and-c-so-that-the-direction-derivative-of-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-

Question Number 65797 by Souvik Ghosh last updated on 04/Aug/19
$${find}\:{the}\:{constant}\:\:{a},{b}\:{and}\:\:{c}\:\:{so} \\$$$${that}\:{the}\:{direction}\:{derivative}\:{of} \\$$$$\Phi={axy}^{\mathrm{2}} +{byz}+{cz}^{\mathrm{2}} {x}^{\mathrm{3}} \:{at}\:\left(\mathrm{1},\mathrm{2},−\mathrm{1}\right) \\$$$${has}\:{a}\:{maximum}\:{of}\:{magnitude} \\$$$$\mathrm{64}\:{jn}\:{a}\:{direction}\:{parallel}\:{to}\:{the} \\$$$${z}\:{axis}. \\$$
Answered by Tanmay chaudhury last updated on 04/Aug/19
$$\bigtriangledown\Phi \\$$$$\left({i}\frac{\partial}{\partial{x}}+{j}\frac{\partial}{\partial{y}}+{k}\frac{\partial}{\partial{z}}\right)\left({axy}^{\mathrm{2}} +{byz}+{cz}^{\mathrm{2}} {x}^{\mathrm{3}} \right) \\$$$${i}\left({ay}^{\mathrm{2}} +\mathrm{3}{x}^{\mathrm{2}} {cz}^{\mathrm{2}} \right)+{j}\left(\mathrm{2}{axy}+{bz}\right)+{k}\left({by}+\mathrm{2}{cx}^{\mathrm{3}} {z}\right) \\$$$$\\$$$$\\$$$$\\$$$$\\$$
Commented by Tanmay chaudhury last updated on 04/Aug/19
$${eait}\:{i}\:{have}\:{corrected}\:{my}\:{error}\:{detected}\:{by}\:{you}\:..{thank}\:{you} \\$$$${wait}\:{iam}\:{trying} \\$$
Commented by Souvik Ghosh last updated on 04/Aug/19
$${but}\:{sir}\:{hear}\: \\$$$$\overset{\rightarrow} {\bigtriangledown}\Phi=\left({ay}^{\mathrm{2}} +\mathrm{3}{cz}^{\mathrm{2}} {x}^{\mathrm{2}} \right){i}+\left(\mathrm{2}{axy}+{bz}\right){j} \\$$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\left({by}+\mathrm{2}{czx}^{\mathrm{3}} \right){k} \\$$
Commented by Tanmay chaudhury last updated on 04/Aug/19
$${yes}\:{you}\:{are}\:{right}…{in}\:{hurry} \\$$