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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 $u .$
$\begin{aligned} \begin{aligned} f (x, y) = 3 x^{2} 鈭� 4 x y + 2 y^{2} P = (0, 0) \end{aligned} \end{aligned}$

Short Answer

Expert verified

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

localid="1650601265652" $鈭� f (P) 脳 u = 0$

Step by step solution

01

Expression of solution

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

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

02

Calculation

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

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

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

$鈭� f (P) 脳 u = 0$

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Most popular questions from this chapter

In Exercises, find the maximum and minimum of the function f subject to the given constraint. In each case explain why the maximum and minimum must both exist.

$f (x, y) = x y when x^{2} + 4 y^{2} = 16$

Describe the meanings of each of the following mathematical expressions:

$鈭� f (x, y, z)$

Extrema: Find the local maxima, local minima, and saddle points of the given functions.

$f (x, y) = 2 x^{2} + y^{2} + y + 5 .$

Prove that if you minimize the square of the distance from the origin to a point (x, y) subject to the constraint $g (x, y) = 0$ , you have minimized the distance from the origin to (x, y) subject to the same constraint.

$Let f (x) be a differentiable function of x, g (y) be a differentiable function of y, and h (x, y) = f (x) + g (y) . Prove that \frac{{鈭�}^{2} h}{鈭� x 鈭� y} = \frac{{鈭�}^{2} h}{鈭� y 鈭� x} .$

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