Consider the following functions. f,(x) = x, f,(x) = x², f,(x) = 5x - 4x² g(x) = c,f,(x) + C2f2(x) + czf3(x) Solve for c,, c,, and c, so that g(x) = 0 on the interval (-o, o). If a nontrivial solution exists, state it. (If only the trivial solution exists, enter the trivial solution {0, 0, 0}.) {C,, C2, C3} = Determine whether f, f, fa are linearly independent on the interval (-0, o). O linearly dependent O linearly independent

Q: (b) Solve for the optimum to the following problem using the Karush-Kuhn Tucker conditions: Optimise…

A: Solve the following using the Karush-Kuhn Tucker conditions: Optimise fx1,x2=x1x2s.t.…

Q: Minimize f(x) = x² + 54 ; using Interval halving method for 2 ≤ x ≤ 6. E= 10-³

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Q: Consider an arbitrary quadratic function f(x) = a.x² + bx +c where a, b, and c are constants with a…

A: We need to find second derivative test to check local maximum and minimum

Q: 2. Find and classify the critical points of the function f (x, y) = 6x2 – 2x³ + 3y? + 6xy

A: The solution is given below in the next step:

Q: +] has the general solution u(x, y) = e"[f(x)+g(y)] where f and g g are arbitrary functions

A: ∂2U∂xdy+∂u∂x=0let U(H.Y)=e-yf(x)+g(y)

Q: particular

A: We find particular solution.

Q: Find the values of x, y and z that correspond to the critical point of the function f(x, y) = 2x?.…

A: The given function is  f(x, y) = 2x2 - 5x - 3y + 5y2 To find the critical points of the function.

Q: Find the critical numbers of the function. g(y) = y2 - 2y + 8

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Q: x" – 2x' + 4x = d(t – 1) x(0) = x'(0) = 0 - -

A: For solving this question we use the table of Laplace transform and properties of Laplace transform…

Q: where y1 = e* is one of the functions in the fundamental solution set. - %3D xy" + (2-2x)y' +(x- 2)y…

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Q: Find the critical point of f (x, y) = 3x² + xy - 5x + y°, and determine its nature.

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Q: Find the critical numbers of the function. y-4 g(v) = y2 - 2y + 8

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Q: ľ Solve the following optimization problem (using second order conditions to confirm your…

A: note :  As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: 14.Show that any separable equation M(x)+N(y)y′=0 is also exact.

A: Given: Mx+Nyy'=0

Q: Find the general solution by using reduction of order method to (x2 + 1)y" – 2xy' + 2y = 0 given…

A:

Q: Find the Quadratic approximation of the function at the given value f(x) = 1n(x + 1)     near a = 0

A: Given  The function is f(x) = ln(x + 1)     near a = 0 .   Formula for Quadratic approximation :-…

Q: Find the values of x, y and z that correspond to the critical point of the function: 23D f(+, у) —…

A: Given  z = f(x, y) = 2x2-6x-3y+3y2+5xy We have to find the critical point of the function.

Q: Use the variation of parameters to find the solution to the IVP. 0 1 et 1 X' (t) = x(t) + ,and X(0)…

A:

Q: Find the quadratic function f (x) = ax2 + bx + c that goes through (0, 1) and has a local minimum at…

A: Given: The quadratic function fx=ax2+bx+cpasses through the point 0,1 and has local minimum at 1,-1,…

Q: Verify that y = h(x) =- (C is an arbitrary constant, x # 0) is a solution of xy' = -y.

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Q: Find all stationary (critical) points of the function g(x, y) = x³ + y³ + 3x² + 3y + 3x + 3xy +1.

A: First calculate the partial derivatives, gx, gy  for g(x,y) . Solve the equations, gx=0 , gy=0…

Q: Suppose that x-y" + axy' + by = 0 has the general solution of the following form: y = C,x³ + C2x³ In…

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Q: Find the particular solution determined by the initial condition. P(X) = 2x ° = 3x )= -6 2/5 - 3x:…

A: This belong to initial value problem of differential equation.first we solve the differential…

Q: Show that y = - In x + 3x - 5 is a solution of y" -=0. Use y = - Inx+3x – 5 to find y' and y". y' =O

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Q: use the Lagrangian method to solve: U=x1^(1/2)x2^(1/2) subject to 50=5x1+10x2

A:

Q: The function f(x) = x^3 + bx^2 + c has a critical point at (-2, 6). Determine the value of b and c.

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Q: x²y = 1+ cx

A: the given equation is x2y=1+cx

Q: Use the IVT to show that e−x2 = x has a solution on (0, 1).

A: Given equation is

Q: Find the particular solution determined by the initial condition. f'(x) = rx) = 2x2/5 - 3x, f(1) =…

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Q: Consider the following where c,, c2, C3 and ca are constants. c,(1 - 4x) + c2(9x + x2 - x) + c3(1 +…

A: Concept:  Say we have few vectors and we write them in matrix form.  If a matrix has no free…

Q: Find the critical point (xo, yo) of the function f(x, y) = 2x2 + 2xy + y? + 2x – 3.

A: Consider both the image for the solution.

Q: Given that y1 = 3x is a solution of æ°y" + xy' - y = x2 on the interval (0, 0), use the reduction of…

A: Concept - used the reduction of order

Q: x² + y² F(x, y) =- - is a homogeneous function of degree

A: For finding degree of homogeneous function, Put x=hx and y=hy

Q: Show solution/process. 1. f(x) = 2x² – 4x + 5 at the point (–1,11) 2. f(x) = v25 – x² at the point…

A: Since there are more than one question, only one question can be answered. The function is given by,…

Q: Solve f, e-+ n² dt. 6.

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Q: Maximize the function f(x,x2), using the Newton Optimization Method. f(x1, x2) = x1x2 + x1 +5x,2 +…

A:

Q: A particular solution of x¨ − 3x˙ + 2x = t2 + 1 is given by

A: given that x''-3x'+2x=t2+1. we need to find the particular solution for the given differential…

Q: Solve the following optimization problem by using Lagrange method: Maximize f(x,x;)=2x, + x, – 3x²…

A: Lagrange Multiplier Method:- This method is used to optimize a function on a region that contained…

Q: Find the critical point of the function f(x,y) = 7x²y² + 4y² + 4x².

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Q: (b) Find the general solution of: = yx² - xy ax Given z = y when x=1, determine the particular…

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Q: Find the critical numbers of the function. g(y) = y-5/y2-3y+15

A: Given function is, g(y) = y-5/y2-3y+15

Q: Let y : R → R be the real-valued function defined on the real line, which is the solution of the…

A: The differential equation is given as Y’  = -xy + x , y(o) = 2 From the type of the problem we can…

Q: The value of Wronskian of two functions y, =7* and y2 =7** is

A:

Q: For the function f(x, y) = 4x³x² + 6x - 7, find the solution from x = 0 to x = 2 with an increment…

A: The numerical solution of the differential equation is the value of the function that satisfies the…

Q: Use variation of parameters to find a particular solution of y'' - Y = 4e-x 1-e-2x Yp (x) =

A: Solution of the differential equation, By using the method of Variation of perameters

Q: Solve the following optimization problem (using second order conditions to confirm your…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: 1 Find the general solution of x'y" +xy' -y= x+1

A:

Q: Find the absolute maximum value of the function f(x) =-z² – 4x – 2 on the interval [3, 3 ]. a) О-2…

A:

Q: Given the function y = 8e-²x² + 7 sec² 5x – 4 ln x³, find y'. Solution: y' = 4(3)x² x³3 +8(−4x)e-²x²…

A: We will find the y' of the given function in the following step.

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: