Vector calculus

Anandiscool

If directional derivative of $\phi = axy^{2}+byz+cz^{2}x^{3}$ at $(1,2,-1)$ has a maximum magnitude of $64$ in the direction parallel to the Z-axis,find a,b,c.
Cool (speak)

**Posted:** Sat Oct 31, 2009 9:57 pm

anik_andrew · Riemann Hypothesis Offline Joined: 27 Dec 2005 Posts: 273

$grad\ f |_M = (4a+3c,4a-b,2b-2c) = V$ . Let $v = (\alpha,\beta,\gamma).$
Then $\frac{\partial f}{\partial v} = \left(\frac{v}{\|v\|}, V\right)$ . If $\frac{\partial f}{\partial v} \to max$ , then $v$ and $V$ are colinear: $v=kV,k\in \mathbb{R}.$
Then $4a+3c=0, 4a-b=0,|2b-2c|=64$ . And we can obtain the answer.

**Posted:** Sun Nov 01, 2009 5:28 am