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Chapter 21: Problem 1

Locate the position of any stationary points of the following functions: $$ f(x, y)=x^{2}+y^{3}-3 y $$

Short Answer

Expert verified
Answer: The stationary points of the given function are at \((0, 1)\) and \((0, -1)\).

Step by step solution

01

Identify the function

We have the function given as: $$ f(x, y) = x^2 + y^3 - 3y $$ The task is to find the stationary points of this function.
02

Calculate partial derivatives

We will now calculate the partial derivatives of the function with respect to x and y. First, the partial derivative with respect to x: $$ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x}\left(x^2 + y^3 - 3y\right) = 2x $$ Now, the partial derivative with respect to y: $$ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y}\left(x^2 + y^3 - 3y\right) = 3y^2 - 3 $$
03

Set partial derivatives to zero

To find the stationary points, we need to set the partial derivatives equal to zero: $$ \begin{cases} 2x = 0 \\ 3y^2 - 3 =0 \end{cases} $$
04

Solve the system of equations

Now, we need to solve the system of equations obtained in Step 3: For the first equation, we have $$ x = 0 $$ For the second equation, $$ 3y^2 - 3 = 0 \Rightarrow y^2 = 1 \Rightarrow y = \pm 1 $$
05

Find stationary points

We have found the values of x and y that make the partial derivatives equal to zero. Let's now list the stationary points of the function: $$ (0, 1), \, (0, -1) $$ So, the function has two stationary points at \((0, 1)\) and \((0, -1)\).

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Most popular questions from this chapter

If \(z=f(x, y)=-11 x+y\) find (a) \(f(2,3)\) (b) \(f(11,1)\).

In each case, given \(z=f(x, y)\), find \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\). (a) \(z=5 x+11 y\) (b) \(z=-7 y-14 x\) (c) \(z=8 x(\) d) \(z=-5 y\) (e) \(z=3 x+8 y-2\) (f) \(z=17-3 x+2 y\) (g) \(z=8\) (h) \(z=8-3 y\) (i) \(z=2 x^{2}-7 y\) (j) \(z=9-3 y^{3}+7 x\) (k) \(z=9-9(x-y) \quad\) (I) \(z=9(x+y+3)\)

$$ \text { If } V=D^{1 / 4} T^{-5 / 6} \text { find } \frac{\partial V}{\partial T} \text { and } \frac{\partial V}{\partial D} $$

Given \(z=f(x, y)=7 x+2 y\) find the output when \(x=8\) and \(y=2\).

If \(f(x, t)=\mathrm{e}^{2 x}\) find \(f(0.5,3)\).
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