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Given f(x,y)=x^3-4xy+2y^2-1 , Using the second derivative test for multi variable functions find...

Question:

Given {eq}f(x,y)=x^3-4xy+2y^2-1 , {/eq} Using the second derivative test for multi variable functions find and classify any critical points. Show all work.

Critical Points:

To determine the critical points we need to find the first and second partial derivatives of the given function and we will solve for value of {eq}x {/eq} and {eq}y {/eq}. And to classify the critcial points note that in second derivative test if {eq}D\left ( a,b \right )> 0 \:and \:f_{xx}> 0 {/eq} then {eq}f(x,y) {/eq} has a loca minimum at (a,b) whereas if {eq}D\left ( a,b \right )> 0 \:and \:f_{xx}< 0 {/eq} then the point {eq}(a,b) {/eq} is a local maximum and finally if {eq}D\left ( a,b \right )< 0 {/eq} then the point is a saddle point and where {eq}D=f_{xx}(x,y)f_{yy}(x,y)-f_{xy}(x,y)^{2} {/eq}

Answer and Explanation:

The partial derivatives of the given function are the following,

{eq}f=x^3-4xy+2y^2-1 {/eq}

{eq}f_{x}=3x^{2}-4y {/eq}

{eq}f_{y}=-4x+4y {/eq}

{eq}f_{xx}=6x \\ f_{yy}=4 {/eq}

{eq}f_{xy}=-4 f_{yx}=-4 {/eq}

Solving for critial points we let {eq}f_{x}=0,f_{y}=0 {/eq}

Then, from

{eq}f_{y}=-4x+4y {/eq}

{eq}0=-4x+4y {/eq}

{eq}x=y\rightarrow (1) {/eq}

Substituting (1) to {eq}f_{x} {/eq}

{eq}f_{x}=3x^{2}-4y {/eq}

{eq}0=3x^{2}-4x {/eq}

{eq}0=x\left ( 3x-4\right ) {/eq}

{eq}x=0,\frac{4}{3} {/eq}

Now when {eq}x=0,y=0 {/eq}

and when {eq}x=\frac{4}{3},\:y=\frac{4}{3} {/eq}

Thus the critical points are {eq}\left ( 0,0 \right ),\left ( \frac{4}{3},\frac{4}{3} \right ) {/eq}

For the critical point {eq}(0,0) {/eq},

{eq}D=6(0)\times (4)-(4)^{2}< 0 {/eq} thus the pooint (0,0) is saddle point

Fro critical point {eq}( \frac{4}{3},\frac{4}{3}) {/eq}

{eq}D=6\left ( \frac{4}{3} \right )\times (4)-(4)^{2}> 0,\:f_{xx}=8 {/eq}

Thus, the point {eq}( \frac{4}{3},\frac{4}{3}) {/eq} is local minimum.


Learn more about this topic:

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Finding Critical Points in Calculus: Function & Graph

from CAHSEE Math Exam: Tutoring Solution

Chapter 8 / Lesson 9
163K

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