Consider the function f(x,y) = 3xe" – x³ – e3y. (a) Show that f(x,y) has exactly one critical point and that f(x, y) has a local maximum value at this point. (b) Does f(x, y) have a global maximum value? Explain.

Q: Find the values of x, y, z at which f (x, y, z) = x2 + 4y2 + 5z2 - 6x + 8y + 3 assumes its minimum…

A: f(x, y, z)=x2+4y2+5z2-6x+8y+3   we have to find all points of x, y, z Taking, ∂f∂x=0, ∂f∂y=0 and…

Q: Find critical point (0,0) of f(x,y) =e^[(-2x^2)+(3y^2)] is a min, max, or saddle point using the…

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Q: The function f..) = x+x +3x- 1+2r,x + 2 defined on R has ohly one stationary point. Show that it is…

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Q: Consider the function g(x, y) = x³ – 4x + ye" – In y. a. Find all critical points of g. At each…

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Q: 1. Ta)Galculate the critical polnts of Hhe function z(x, y) = K-1} + (q-3) - 5, subjet to %3D X +2y…

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Q: 3. Let f(r, y)- e (a) Verify that (0, 0) is a critical point. (b) Is this critical point a local…

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Q: Find the point where the maximum value of f(x, y) = xy subject to y = 33 – x² occurs. (x, y, z) =

A: (x,y,z) = (√11,  22,  22√11)  The detailed solution is as follows below:

Q: Suppose that f(x, y) = x? – zy + y² – 3z + 3y, with domain D constrained by the lines y = z, y = 0 ,…

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Q: fc is a critical point of F(x), when the is a local maximum or minimum at c? Explain

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Q: 2. The function f(x, y) = S* cos(t²)dt has exactly one critical point in the domain 0 < x < 2,0 < y…

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Q: Show that f(x, y) = x^2 + y^2 − 2xy + 4 has infinite number of critical points and that D = 0 at…

A: Given that      f(x,y) = x2+y2 -2xy+4 has infinite number of critical points and D= 0 at each one.…

Q: The curved surface of a rod of length L is perfectly insulated against the flow of heat. The rod…

A: .

Q: 8. Let f(x,y) = x² – 8x + 2y? + 4y +7. - a. Find and classify all critical points of f. b. Does f…

A: I have used first derivative rule to find critical point and for classification I used second order…

Q: Find the local minimum value of f(x, y) = 4 + a" + yl – 11ry.

A: Use double derivatives test to find local maximum or minima.

Q: f z = f (x, y) is a function that admits second continuous partial derivatives such that: ∇f (x, y)…

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Q: Classify all the critical point(s) of f(x, y) = 22 + 6x? – 3y3 – 150.x. Find maximum and minimum…

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Q: Suppose that f(x, y) = x² + y² at which 0 < x, y and 2x + 9y < 8. 1. Absolute minimum of f(x, y) is…

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Q: Let f(x, y) = cos(x)cos(y). Find all critical points of f which lie in the square {(x, y) ∈ R2 : −1…

A: f(x,y) = = cos(x)cos(y)At critical points, partial derivatives of f (x,y) w.r.t x and y are…

Q: Consider the finction f(x, y) = 6y² + 4x² – 12xy- 4y. Then: (1,1) is a critical point and f has a…

A: The given function is f(x,y)=6y2+4x2-12xy-4y

Q: Suppose that f(x, y) = x² + y? at which 0 < x, y and 5x + 6y < 8. 1. Absolute minimum of f(x, y) is…

A: This question is related to linear programming, we will solve it using given information.

Q: Suppose that f(x, y) = x² – xy + y? – læ + ly with D = {(x, y) | 0 < y < x < 1}. - - 1. The critical…

A: Given: f(x,y)=x2-xy+y2-x+y. (1) Partially differentiate the function with respect to x and y…

Q: Find A and B so that f(x, y) = x2 + Ax + y² + B has a local minimum at the point (4, 0), with…

A: Given function of two variables is f(x,y)=x2+Ax+y2+B it has local minimum point at (4,0) and minima…

Q: Let f (x, y) = y^2x − yx^2 + xy.  (a) Show that the critical points (x, y) satisfy the equations y(y…

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Q: (a) Find the critical point of f(x, y) = (4x2 y) (x² + 4y). (x,y)%3D (b) Find the discriminant, D(x,…

A: f(x, y) = 4x2-yx2+4y

Q: Define the function f: R? → R by f (x, y) = 2xy² – y² – 2x². a) Find the critical points of f. b)…

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Q: f(x, y) attains a global maximum at a point in the interior of R (that is, at a point in R that is…

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Q: Suppose that f(x, y, z) = 2x + 5y + 4z at which a2 + y? + z2 < 12. 1. Absolute minimum of f(x, y, z)…

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Q: Show that f(x,y) = 1-x* - y* has a relative maximum, even though D in the theorem is 0. The critical…

A: Given : f(x, y) = 1 - x⁴ - y⁴

Q: Suppose that f(x, y) = x² + y² at which 0 < x, y and 4x + 7y < 9. 1. Absolute minimum of f(x, y) is…

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Q: J(X) = -3|X – 5| + 2 1. Why does J(X) have a critical point at X = 5? Explain. 2. Why does J(X)…

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Q: Consider the function f(x, y) = y6 - 3x²y + 3x² Compute the stationary points of the function f(x,…

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Q: Consider the function f(x, y) = x³y+ 9x² – y, defined on the set D = {(x, y) | x² + y² < 100}. (a)…

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Q: Suppose that f(x, y) = x² – xy+ y² – x + y with D = {(x, y) |0 < y< x < 1}. 1. The critical point of…

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Q: Let f(x, y) = x* + y* – 4xy. (a) Find all critical points of f. (b) For each critical point of f,…

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Q: Show that the Second Derivative Test is inconclusive when applied to the following function at…

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Q: Suppose that the function f(x, y) has a critical point at (2, y) = (a, b). Furthermore, fæz (a, b)…

A: To decide the required conclusion, we use the second derivative test for maxima and mimima.

Q: Suppose that f(x, y, z) = 4x + 4y + 3z at which 0 < x, y, z < 5. 1. Absolute minimum of f(x, y, z)…

A: We have to find the absolute minima and maxima.

Q: Let f be a function that admits second continuous partial derivatives on which is know what: ∇f(x,…

A: Given : ∆f(x ,y) = ( 3x2 - 300 , 3y2 - 75) Fxx = 6x , fyy = 6y

Q: Using these equations 8x+2xlambda=0 4y+2ylambda=2 x^2+y^2=2y Find x, y, critical point,…

A: 8x+2xlambda=0 4y+2ylambda=2 x^2+y^2=2y

Q: Suppose that f(*, 3) = x? – xy +y? – 5x + 5y with D = {(*, y) | 0 <y <* < 5}. 1. The critical point…

A: Given that a function f(x,y) and we find critical points and absolute maximum and minimum. Here i…

Q: Compute the maximum value of the utility function U(x,y, z) = xyz, subject to x+ 3y+4z 60. %3D

A: To find: Maximum of the function U(x,y,z)=xyz subject to x+3y+4z=60.

Q: 1)Consider the function f(x, y) = x° + 2xy + y? – 5x b) Calculate the second-order partial…

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Q: Consider the function f(x, y) = -x²y² + 4y¹ + 4x² Compute the stationary points of the function f(x,…

A: Given function is fx,y=−x2y2+4y4+4x2. We have to compute the stationary points o the function fx,y…

Q: Find the point at which the function f(r,y) = 2 in z+2 In y attains a maximum subject to the con-…

A: Maxima or minima: For any function f(x), 1) Evaluate f'(x)=0 for critical point. 2) Evaluate f''(x)…

Q: Show that (x, y, z) = xy + y² + 22 has no local extrema by showing that its unique critical point is…

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Q: Let f(x, y) = (x² +y²)e=*-v°. Find and classify all critical points. %3D

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Q: (a) Find the first partial derivatives of f(x, y) = cos(2.ry) – e³y+2x. (b) Evaluate (0,0) and (T,…

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Q: Consider the function f(x, y) = x²(y + 1)² + k(x + 1)²y². (a) For what value(s) of k is the point…

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Q: (b) Form the partial equation by eliminating function f from z = f(x?

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Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.