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CalculusCalculus: Early TranscendentalsUse a graphing device as in Example 4 (or Newton’s method or solve numerically using a calculator or computer) to find the critical points of f correct to three decimal places. Then classify the critical points and find the highest or lowest points on the graph, if any. 30. f ( x , y ) = 20 e − x 2 − y 2 sin 3 x cos 3 y , | x | ≤ 1, | y | ≤ 1BuyFind*arrow_forward*

8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781285741550

Chapter 14.7, Problem 30E

Textbook Problem

Use a graphing device as in Example 4 (or Newton’s method or solve numerically using a calculator or computer) to find the critical points of *f* correct to three decimal places. Then classify the critical points and find the highest or lowest points on the graph, if any.

**30.**
*x*| ≤ 1, |*y*| ≤ 1

Expert Solution

To determine

**To find:** The critical points of the function

**Result used:**

**Second Derivative Test:**

“Suppose the second partial derivatives of *f* are continuous on a disk with center
*f*).

Let

**Given:**

The function is,

**Calculation:**

Take the partial derivative in the given function with respect to *x* and obtain

Thus,

Take the partial derivative in the given function with respect to *y* and obtain

Thus,

Solve the equations (1) and (2) and find the values of *x* and *y*.

From the equation (1),

From the result, there are two possibilities such either

Suppose,

Suppose

Find the value of

Use online graphing calculator and draw the graph of the equation

From Figure 1 it can be observed that the values of

From equation (2),

From the result, there are two possibilities such either

Suppose

Consider

Find the value of

Use the online graphing calculator and draw the graph of the equation

From Figure 1 it can be observed that the values of

Therefore, the critical points of the function

Obtain the second derivative,

Take the partial derivative of the equation (1) with respect to *x* and obtain

Hence,

Take the partial derivative of the equation (2) with respect to *y* and obtain

Hence,

Take the partial derivative of the equation (1) with respect to *y* and obtain

Hence,

Use second derivative test and obtain the value of *D* for the critical point

Since

Substitute

Similarly, obtain the value of *D* for the critical point

Since,

Substitute

Obtain the value of *D* for the critical point

Since,

Substitute

Calculus: Early Transcendentals

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(b) Evaluate the...Ch. 14.6 - (a) Find the gradient of f. (b) Evaluate the...Ch. 14.6 - (a) Find the gradient of f. (b) Evaluate the...Ch. 14.6 - (a) Find the gradient of f. (b) Evaluate the...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Use the figure to estimate Du, f(2, 2).Ch. 14.6 - Find the directional derivative of f(x, y) = xy at...Ch. 14.6 - Find the directional derivative of f(x, y, z) =...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - (a) Show that a differentiable function f...Ch. 14.6 - Find the directions in which the directional...Ch. 14.6 - Find all points at which the direction of fastest...Ch. 14.6 - Near a buoy, the depth of a lake at the point with...Ch. 14.6 - The temperature T in a metal ball is inversely...Ch. 14.6 - The temperature at a point (x, y, z) is given by...Ch. 14.6 - Suppose that over a certain region of space the...Ch. 14.6 - Suppose you are climbing a hill whose shape is...Ch. 14.6 - Let f be a function of two variables that has...Ch. 14.6 - Shown is a topographic map of Blue River Pine...Ch. 14.6 - Show that the operation of taking the gradient of...Ch. 14.6 - Sketch the gradient vector f(4. 6) for the...Ch. 14.6 - The second directional derivative of f(x, y) is...Ch. 14.6 - (a) If u = a, b is a unit vector and f has...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Use a computer to graph the surface, the tangent...Ch. 14.6 - Use a computer to graph the surface, the tangent...Ch. 14.6 - If f(x, y) = xy, find the gradient vector f(3, 2)...Ch. 14.6 - If g(x, y) = x2 + y2 4x, find the gradient vector...Ch. 14.6 - Show that the equation of the tangent plane to the...Ch. 14.6 - Find the equation of the tangent plane to the...Ch. 14.6 - Show that the equation of the tangent plane to the...Ch. 14.6 - At what point on the ellipsoid x2 + y2 + 2z2 = 1...Ch. 14.6 - Are there any points on the hyperboloid x2 y2 z2...Ch. 14.6 - Show that the ellipsoid 3x2 + 2y2 + z2 = 9 and the...Ch. 14.6 - Show that every plane that is tangent to the cone...Ch. 14.6 - Show that every normal line to the sphere x2 + y2...Ch. 14.6 - Where does the normal line to the paraboloid z =...Ch. 14.6 - At what points does the normal line through the...Ch. 14.6 - Show that the sum of the x-, y-, and z-intercepts...Ch. 14.6 - Show that the pyramids cut off from the first...Ch. 14.6 - Find parametric equations tor the tangent line to...Ch. 14.6 - (a) The plane y + z = 3 intersects the cylinder x2...Ch. 14.6 - Where does the helix r(t) = cos t, sin t, t...Ch. 14.6 - The helix r(t) = cos(t/2), sin(t/2), t) intersects...Ch. 14.6 - (a) Two surfaces are called orthogonal at a point...Ch. 14.6 - (a) Show that the function f(x, y) = xy3 is...Ch. 14.6 - Suppose that the directional derivatives of f(x,...Ch. 14.6 - Show that if z = f(x, y) is differentiable at x0 =...Ch. 14.7 - Suppose (1, 1) is a critical point of a function f...Ch. 14.7 - Suppose (0, 2) is a critical point of a function y...Ch. 14.7 - Use the level curves in the figure to predict the...Ch. 14.7 - Use the level curves in the figure to predict the...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Show that f(x, y) = x2 + 4y2 4xy + 2 has an...Ch. 14.7 - Show that f(x,y)=x2yex2y2 has maximum values at...Ch. 14.7 - Use a graph or level curves or both to estimate...Ch. 14.7 - Use a graph or level curves or both to estimate...Ch. 14.7 - Use a graph or level curves or both to estimate...Ch. 14.7 - Use a graph or level curves or both to estimate...Ch. 14.7 - Use a graphing device as in Example 4 (or Newtons...Ch. 14.7 - Use a graphing device as in Example 4 (or Newtons...Ch. 14.7 - Use a graphing device as in Example 4 (or Newtons...Ch. 14.7 - Use a graphing device as in Example 4 (or Newtons...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - For functions of one variable it is impossible for...Ch. 14.7 - If a function of one variable is continuous on an...Ch. 14.7 - Find the shortest distance from the |point (2, 0,...Ch. 14.7 - Find the point on the plane x 2y + 3z = 6 that is...Ch. 14.7 - Find the points on the cone z2 = x2 + y2 that are...Ch. 14.7 - Find the points on the surface y2 = 9 + xz that...Ch. 14.7 - Find three positive numbers whose sum is 100 and...Ch. 14.7 - Find three positive numbers whose sum is 12 and...Ch. 14.7 - Find the maximum volume of a rectangular box that...Ch. 14.7 - Find the dimensions of the box with volume 1000...Ch. 14.7 - Find the volume of the largest rectangular box in...Ch. 14.7 - Find the dimensions of the rectangular box with...Ch. 14.7 - Find the dimensions of a rectangular box of...Ch. 14.7 - The base of an aquarium with given volume V is...Ch. 14.7 - A cardboard box without a lid is to have a volume...Ch. 14.7 - A rectangular building is being designed to...Ch. 14.7 - If the length of the diagonal of a rectangular box...Ch. 14.7 - A model for the yield Y of an agricultural crop as...Ch. 14.7 - The Shannon index (sometimes called the...Ch. 14.7 - Three alleles (alternative versions of a gene) A,...Ch. 14.7 - Suppose that a scientist has reason to believe...Ch. 14.7 - Find an equation of the plane that passes through...Ch. 14.8 - Pictured are a contour map of f and a curve with...Ch. 14.8 - (a) Use a graphing calculator or computer to graph...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - The method of Lagrange multipliers assumes that...Ch. 14.8 - Find the minimum value of f(x, y, z) = x2 + 2y2 +...Ch. 14.8 - Find the extreme values of f subject to both...Ch. 14.8 - Find the extreme values of f subject to both...Ch. 14.8 - Find the extreme values of f subject to both...Ch. 14.8 - Find the extreme values of f subject to both...Ch. 14.8 - Find the extreme values of f on the region...Ch. 14.8 - Find the extreme values of f on the region...Ch. 14.8 - Find the extreme values of f on the region...Ch. 14.8 - Consider the problem of maximizing the function...Ch. 14.8 - Consider the problem of minimizing the function...Ch. 14.8 - The total production P of a certain product...Ch. 14.8 - Referring to Exercise 27, we now suppose that the...Ch. 14.8 - Use Lagrange multipliers to prove that the...Ch. 14.8 - Use Lagrange multipliers to prove that the...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Find the maximum and minimum volumes of a...Ch. 14.8 - The plane x + y + 2z = 2 intersects the paraboloid...Ch. 14.8 - The plane 4x 3y + 8z = 5 intersects the cone z2 =...Ch. 14.8 - (a) Find the maximum value of...Ch. 14.8 - (a) Maximize i=1nxiyi subject to the constraints...Ch. 14 - (a) What is a function of two variables? (b)...Ch. 14 - What is a function of three variables? How can you...Ch. 14 - What does lim(x,y)(a,b)f(x,y)=L mean? How can you...Ch. 14 - (a) What does it mean to say that f is continuous...Ch. 14 - (a) Write expressions for the partial derivatives...Ch. 14 - What does Clairauts Theorem say?Ch. 14 - How do you find a tangent plane to each of the...Ch. 14 - Define the linearization of f at (a, b). What is...Ch. 14 - (a) What does it mean to say that f is...Ch. 14 - If z = f(x, y), what arc the differentials dx, dy,...Ch. 14 - State the Chain Rule for the case where z = f(x,...Ch. 14 - If z is defined implicitly as a function of x and...Ch. 14 - (a) Write an expression as a limit for the...Ch. 14 - (a) Define the gradient vector f for a function f...Ch. 14 - What do the following statements mean? (a) f has a...Ch. 14 - (a) If f has a local maximum at (a, b), what can...Ch. 14 - State the Second Derivatives Test.Ch. 14 - (a) What is a closed set in 2? What is a bounded...Ch. 14 - Explain how the method of Lagrange multipliers...Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Determine whether the statement is true or false....Ch. 14 - Find and sketch the domain of the function. 1....Ch. 14 - Find and sketch the domain of the function. 2....Ch. 14 - Sketch the graph of the function. 3. f(x, y) = 1 ...Ch. 14 - Sketch the graph of the function. 4. f(x, y) = x2...Ch. 14 - Sketch several level curves of the function. 5....Ch. 14 - Sketch several level curves of the function. 6....Ch. 14 - Make a rough sketch of a contour map for the...Ch. 14 - The contour map of a function f is shown, (a)...Ch. 14 - Evaluate the limit or show that it does not exist....Ch. 14 - Evaluate the limit or show that it does not exist....Ch. 14 - A metal plate is situated in the xy-plane and...Ch. 14 - Find a linear approximation to the temperature...Ch. 14 - Find the first partial derivatives. 13. f(x, y) =...Ch. 14 - Find the first partial derivatives. 14....Ch. 14 - Find the first partial derivatives. 15. F(, ) = 2...Ch. 14 - Find the first partial derivatives. 16. G(x, y, z)...Ch. 14 - Find the first partial derivatives. 17....Ch. 14 - The speed of sound traveling through ocean water...Ch. 14 - Find all second partial derivatives of f. 19. f(x,...Ch. 14 - Find all second partial derivatives of f. 20. z =...Ch. 14 - Find all second partial derivatives of f. 21. f(x,...Ch. 14 - Find all second partial derivatives of f. 22. v =...Ch. 14 - If z = xy + xey/x show that xzx+yzy=xy+z.Ch. 14 - If z = sin(x + sin t), show that zx2zxt=zt2zx2Ch. 14 - Find equations of (a) the tangent plane and (b)...Ch. 14 - Find equations of (a) the tangent plane and (b)...Ch. 14 - Find equations of (a) the tangent plane and (b)...Ch. 14 - Find equations of (a) the tangent plane and (b)...Ch. 14 - Find equations of (a) the tangent plane and (b)...Ch. 14 - Use a computer to graph the surface z = x2 + y4...Ch. 14 - Find the points on the hyperboloid x2 + 4y2 z2 =...Ch. 14 - Find du if u = ln(1 + se2t).Ch. 14 - Find the linear approximation of the function...Ch. 14 - The two legs of a right triangle are measured as 5...Ch. 14 - If u = x2y3 + z4, where x = p + 3p2, y = pep, and...Ch. 14 - If v = x2sin y + yexy, where x = s + 2t and y =...Ch. 14 - Suppose z = f(x, y), where x = g(s, t), y = h(s,...Ch. 14 - Use a tree diagram to write out the Chain Rule for...Ch. 14 - If z = y + f(x2 y2), where f is differentiable,...Ch. 14 - The length x of a side of a triangle is increasing...Ch. 14 - If z =f(u, v), where u = xy, v = y/x, and f has...Ch. 14 - If cos(xyz) = 1 + .x2y2 + z2, find zx and zy.Ch. 14 - Find the gradient of the function f(x,y,z)=x2eyz2.Ch. 14 - (a) When is the directional derivative of f a...Ch. 14 - Find the directional derivative of f at the given...Ch. 14 - Find the directional derivative of f at the given...Ch. 14 - Find the maximum rate of change of f(x,y)=x2y+y at...Ch. 14 - Find the direction in which f(x, y, z) =...Ch. 14 - The contour map shows wind speed in knots during...Ch. 14 - Find parametric equations of the tangent line at...Ch. 14 - Find the local maximum and minimum values and...Ch. 14 - Find the local maximum and minimum values and...Ch. 14 - Find the local maximum and minimum values and...Ch. 14 - Find the local maximum and minimum values and...Ch. 14 - Find the absolute maximum and minimum values of f...Ch. 14 - Find the absolute maximum and minimum values of f...Ch. 14 - Use a graph or level curves or both to estimate...Ch. 14 - Use a graphing calculator or computer (or Newtons...Ch. 14 - Use Lagrange multipliers to find the maximum and...Ch. 14 - Use Lagrange multipliers to find the maximum and...Ch. 14 - Use Lagrange multipliers to find the maximum and...Ch. 14 - Use Lagrange multipliers to find the maximum and...Ch. 14 - Find the points on the surface xy2z3= 2 that are...Ch. 14 - A package in the shape of a rectangular box can be...Ch. 14 - A pentagon is formed by placing an isosceles...Ch. 14 - A rectangle with length L and width W is cut into...Ch. 14 - Marine biologists have determined that when a...Ch. 14 - A long piece of galvanized sheet metal with width...Ch. 14 - For what values of the number r is the function...Ch. 14 - Suppose f is a differentiable function of one...Ch. 14 - (a) Newton's method for approximating a root of an...Ch. 14 - If the ellipse x2/a2 + y2/b2 = 1 is to enclose the...Ch. 14 - Show that the maximum value of the function...

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