Let be the vector field

Answer by explaining as best you can.
a. Can there be a flux curve in a horizontal plane?
b. Can there be a flow curve in an ellipsoid?
c. On which surfaces are the flow curves mounted?
d. What kind of curves are flow curves?

Verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution.

d^2y/ dx^2 − 6 dy/dx + 9y = 0;    y = c₁e^3x + c₂xe^3x  When y = c₁e^3x + c₂xe^3x,

Thus, in terms of x,

2) In this problem, y = c₁e^x + c₂e^−x is a two-parameter family of solutions of the second-order DE

y'' − y = 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions.

y(−1) = 7,    y'(−1) = −7

y =

Find the extremum of​ f(x,y) subject to the given​ constraint, and state whether it is a maximum or a minimum. f(x,y)=(4x^2)+(3y^2)-4xy; x+y=22.Find the Lagrange function ​F(x,y,lambda​).Find the partial derivatives.

task one: Erika’s equations

1. The point (1, 5) is on the curve: y = ax2 + bx + c. This point gives the linear equation: 5 = a + b + c.

A second point on the curve, (2, 10) gives the linear equation 10=4a+2b+c

A student called Erika thinks that the point (2, 19) is also on the curve.

a. Use the point (2, 19) to write the third equation.

b. Attempt to solve this system of three linear equations. Explain the meaning of your answer in geometrical terms.

2. Erika realises that she has made an error and that the third point should be (3, 19) not (2, 19) as in question 1 above.

• Rewrite the third equation and solve the new system of three linear equations.

• Write down the quadratic function.

3. Suppose that the third point is written as (3, t). Find all values for t that will change the quadratic function y = ax2 + bx + c into a linear function.

(a) Find the critical numbers of the function f(x) = x⁸(x − 1)⁷.

(b) What does the Second Derivative Test tell you about the behavior of f at these critical numbers?

(c) What does the First Derivative Test tell you? (Enter your answers from smallest to largest x value.)

Consider the following function:
f (x , y , z ) = x 2 + y 2 + z 2 − x y − y z + x + z

1. (a) This function has one critical point. Find it.

2. (b) Compute the Hessian of f , and use it to determine whether the critical point is a local man, local min, or neither?

3. (c) Is the critical point a global max, global min, or neither? Justify your answer.

1. Find the absolute minimum and maximum value of f(x) = x4 − 18x 2 + 7 (in coordinate form) on [-1,4]

2. If f(x) = x3 − 6x 2 − 15x + 3 discuss whether there are any absolute minima or maxima on the interval (2,∞)

show work please

Use Lagrange multipliers to find the maximum production level when the total cost of labor (at $111 per unit) and capital (at $50 per unit) is limited to $250,000, where P is the production function, x is the number of units of labor, and y is the number of units of capital. (Round your answer to the nearest whole number.) (Please use the numbers given I've followed other 'solutions' and keep getting the wrong answer, I just want to see that the method used in response gets the correct answer and how.)

P(x, y) = 100x^0.25y^0.75

find the point lying on the intersection of the plane, x + (1/4)y + (1/3)z = 0 and the sphere x 2 + y 2 + z 2 = 25 with the largest z-coordinate. (x,y,z)=(_)

how can u=you tell if a serie converge or diverge ?

ƒ(x,y)= x³ + 3xy² - 15x + y³ - 15y

For this question i need to calculate the critical points and all local minima, local maxima and saddle points. How should this be done?

1) Let f(x) be a continuous, everywhere differentiable function and g(x) be its derivative. If f(c) = n and g(c) = d, write the equation of the tangent line at x = c using only the variables y, x, c, n, and d. You may use point-slope or slope-intercept but do not introduce more variables.

2) Let f(x) be a continuous, everywhere differentiable function. What kind information does f'(x) provide regarding f(x)?

3) Let f(x) be a continuous, everywhere differentiable function. What kind information does f''(x) provide regarding f(x)?

4) Let f(x) be a continuous, everywhere differentiable function. What kind information does f''(x) provide regarding f'(x)?

5) Let h(x) be a continuous function such that h(a) = m and h'(a) = 0. Is there enough evidence to conclude the point (a, m) must be a maximum or a minimum? Explain.

(1 point) Suppose that f(x)=(12−2x)e^x.

(A) List all the critical values of f(x). Note: If there are no critical values, enter 'NONE'.

(B) Use interval notation to indicate where f(x) is increasing. Note: Use 'INF' for ∞, '-INF' for −∞, and use 'U' for the union symbol. Increasing:

(C) Use interval notation to indicate where f(x) is decreasing. Decreasing:

(D) List the x values of all local maxima of f(x). If there are no local maxima, enter 'NONE'. x values of local maximums =

(E) List the x values of all local minima of f(x). If there are no local minima, enter 'NONE'. x values of local minimums =

(F) Use interval notation to indicate where f(x) is concave up. Concave up:

(G) Use interval notation to indicate where f(x) is concave down. Concave down:

(H) List the x values of all the inflection points of f. If there are no inflection points, enter 'NONE'. x values of inflection points = (I) Use all of the preceding information to sketch a graph of f. Include all vertical and/or horizontal asymptotes. When you're finished, enter a "1" in the box below.

A spherical ball was completely immersed into an inverted right circular cone full of water. After the ball was removed it was found out that the water surface had dropped 6 cm. below the top of the cone. The diameter of the cone is 12 cm. and its altitude is 36 cm. Determine the wet area (in cm^2) when the cone is full.