Chapter 12: Q. 40 (page 954)

In exercise $39$ - $42$ , show that the directional derivative of the given function at the specified point $P$ is zero for every unit vector $u .$
$\begin{aligned} \begin{aligned} f (x, y) = 3 x^{2} - 4 x y + 2 y^{2} P = (0, 0) \end{aligned} \end{aligned}$

Short Answer

Expert verified

Directional derivation of function at point $P$ with directional unit vector is given by,

localid="1650601265652" $\nabla f (P) \times u = 0$

Step by step solution

Expression of solution

We have given is $f (x, y) = 3 x^{2} - 4 x y - 2 y^{2}$ at specified point $P$ $= (0, 0)$ with unit vector $u = (1, 1)$

localid="1650601329209" $\begin{aligned} \nabla f (P) \times u = \nabla f (0, 0) \times u = ({\frac{d f}{d x}|}_{(0, 0)} i + {\frac{d f}{d y}|}_{(0, 0)} j) \times (\frac{i + j}{\sqrt{i^{2} + j^{2}}}) \end{aligned}$

Calculation

$\begin{array}{r} \nabla f (P) \times u = [(6 x - 4 y)_{(0, 0)} i + (- 4 x - 4 y)_{(0, 0)} j] \times (\frac{i + j}{\sqrt{2}}) \end{array}$

$\begin{array}{r} = [(0 - 0) i + (0 - 0) j] \times (\frac{i + j}{\sqrt{2}}) \end{array}$

$= [0 i + 0 j] \times (\frac{i + j}{\sqrt{2}})$

$\nabla f (P) \times u = 0$

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In exercise $39$ - $42$ , show that the directional derivative of the given function at the specified point $P$ is zero for every unit vector $u .$
$\begin{aligned} \begin{aligned} f (x, y) = 3 x^{2} - 4 x y + 2 y^{2} P = (0, 0) \end{aligned} \end{aligned}$

Short Answer

Step by step solution

Expression of solution

Calculation

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In exercise 39-42, show that the directional derivative of the given function at the specified point Pis zero for every unit vector u.f(x,y)=3x2−4xy+2y2P=(0,0)

Short Answer

Step by step solution

Expression of solution

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Pure Maths

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Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

In exercise $39$ - $42$ , show that the directional derivative of the given function at the specified point $P$ is zero for every unit vector $u .$
$\begin{aligned} \begin{aligned} f (x, y) = 3 x^{2} - 4 x y + 2 y^{2} P = (0, 0) \end{aligned} \end{aligned}$