Answer:
[tex]grad(f(x,y,z))=\frac{\partial f}{\partial x}\widehat{i}+\frac{\partial f}{\partial y}\widehat{j}+\frac{\partial f}{\partial z}\widehat{k}\\\\[/tex]
Applying values we get
[tex]grad(f(x,y,z))=2x\widehat{i}+4y\widehat{j}-6z\widehat{k}[/tex]
b) The directional derivative in the direction of
[tex]\overrightarrow{u}=\widehat{i}+2\widehat{j}+3\widehat{k}[/tex] is given by
[tex]\overrightarrow{\triangledown f}.\frac{\overrightarrow{u}}{|u|}[/tex]
Applying values we get the directional derivative equals
[tex](2x\widehat{i}+4y\widehat{j}-6z\widehat{k}).(\widehat{i}+2\widehat{j}+3\widehat{k})\times \frac{1}{\sqrt{14}}\\\\=\frac{2x+8y-18z}{\sqrt{14}}[/tex]
Thus value at [tex]P_{o}=(1,1,1)=-2.13[/tex]
c)
The direction of rate of maximum increase at [tex]P_{o}=(1,1,1)[/tex] is given by
[tex]\triangledown \overrightarrow{f}(1,1,1)=2\widehat{i}+4\widehat{k}-6\widehat{k}[/tex]
