Lesson Finding the minimum of a function defined on a curve in a coordinate plane
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<H2>Finding the minimum of a function defined on a curve in a coordinate plane</H2> <H3>Problem 1</H3>The positive variables x and y are such that {{{x^4y}}} = 32. A function z is defined by z = {{{x^2 + y}}}. Find the values of x and y that give z a stationary value and show that this value of z is a minimum. <B>Solution</B> <pre> {{{x^4y = 32}}} --> {{{y = 32x^(-4)}}} {{{z = x^2 + y}}} = {{{x^2 + 32x^(-4)}}} (1) {{{dz/dx}}} = {{{2x - 128x^(-5)}}} (2) {{{d^2z/dx^2}}} = {{{2 + 640x^(-6)}}} (3) The stationary point is where the derivative is zero. {{{2x-128x^(-5)}}} = {{{0}}} {{{2x}}} = {{{128x^(-5)}}} {{{x^6}}} = {{{128/2 = 64}}} x = 2 (actually, x = +/- 2, but since we consider everything in positive numbers, we take x = 2). At the stationary point, {{{x = 2}}} and {{{y}}} = {{{32/x^4}}} = {{{32/16}}} = {{{2}}} The stationary point is a minimum if the second derivative at the point is positive; or it is a maximum if that derivative is negative. At x = 2, the second derivative is OBVIOULSLY positive (it is clear without any calculations) So the stationary point is a minimum. ANSWERS: z has a stationary point that is a minimum when x = 2 and y = 2. To make this result more visible and visually verifiable, I prepared a plot below. {{{graph( 400, 400, -5, 5, -5, 20, x^2 + 32x^(-4) )}}} Plot z = {{{x^2}}} + {{{32x^(-4)}}} (see formula (1) </pre> My other lessons on Calculus word problems at this site are - <A HREF=https://www.algebra.com/algebra/homework/word/misc/A-ladder-foot-slides-on-the-ground-txt.lesson>A ladder foot slides on the ground</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Finding-rate-of-change-of-some-processes.lesson>Finding rate of change of some processes</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Find-the-derivative-of-a-function-defined-by-complicated-expression.lesson>Find the derivative of a function defined by complicated expression</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Taking-derivative-of-a-function-which-is-defined-implicitly.lesson>Taking derivative of a function, which is defined implicitly</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Find-the-derivative-of-a-function-satisfying-given-functional-equation.lesson>Find the derivative for a function satisfying given functional equation</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/09-add-Find-the-Range-of-f%28x%29-=-%285cos%28x%29%29-div-%28x-%2B-1%29%29cos%28x%29.lesson>Find the range of f(x) = (5*cos(x))/(x + 1)), x >=0</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/A-tricky-Calculus-problem-on-derivative-and-anti-derivative.lesson>A tricky Calculus problem on derivative and anti-derivative</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Couple-of-non-standard-Calculus-problems.lesson>Couple of non-standard Calculus problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Tricky-solution-to-find-the-maximum-of-a-function-defined-by-a-complicated-expression.lesson>Tricky solution to find the maximum of a function defined by a complicated expression</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Maximize-the-area-of-a-trapezoid.lesson>Maximize the area of a trapezoid </A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Maximmize-the-volume-of-an-open-box.lesson>Maximize the volume of an open box</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Minimize-surface-area-of-a-rectangular-prizm-with-the-given-volume.lesson>Minimize surface area of a rectangular box with the given volume</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Minimize-the-cost-of-an-aquarium-with-the-given-volume.lesson>Minimize the cost of an aquarium with the given volume</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Minimize-surface-area-of-a-conic-paper-cup-with-the-given-volume.lesson>Minimize surface area of a conical paper cup with the given volume</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Find-the-volume-of-a-solid-body-obtained-by-rotation-the-area-about-an-axis.lesson>Find the volume of a solid obtained by rotation of some plane shape about an axis</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Finding-the-volume-of-a-solid-body-mentally.lesson>Finding the volume of a solid body mentally</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/OVERVIEW-of-my-lessons-on-Calculus-word-problems.lesson>OVERVIEW of my lessons on Calculus word problems</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
