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- Find the critical point of the function f(x,y)=3e^x−2xe^y. c=Use the Second Derivative Test to determine whether it isA. test failsB. a local minimumC. a local maximumD. a saddle pointIf z = f (x, y) is a function that admits second continuous partial derivatives suchthat image1 A critical point of f that generates a maximum point is: image2Two curves are orthogonal if their tangent lines areperpendicular at each point of intersection. Show that the givenfamilies of curves are orthogonal trajectories of each other,that is, every curve in one family is orthogonal to every curvein the other family. Sketch both families of curves on the same axes. y = cx 2, x 2+ 2 y 2 = k
- Demonstrate the use of the method with reflections on the use of numerical methods, find the minimum for the function below: F(x,y)= Ax^2 - Bxy- cy^2= x -y (Xo=4, Yo=4) A=2 B=-2 C=1 Identify the minimum again using the Newton’s method with dynamic . However, use this time numerical derivatives instead of . When using numerical derivatives, only one of the constants is being varied as with partial derivatives. Apply in this case the forward numerical derivative, . Here equals some very small number. For each step , solve first the and optimal using the condition . When taking the derivative of , please remember to consider the inner derivatives for each of the coordinate axes that results as dot product with the main function. In this work it is enough that only the second term in the dot product is analyzed using numerical derivatives. Thus, the function takes the form .7) Determine the x - coordinate of the local maximum of the curve y = (2x-1)^2 (3x+4) in the interval [-1.5, 1].Assume the second derivatives of ƒ are continuous throughout the xy-plane and ƒx(0, 0) = ƒy(0, 0) = 0. Use the given information and the Second Derivative Test to determine whether ƒ has a local minimum, a local maximum, or a saddle point at (0, 0), or state that the test is inconclusive. ƒxx(0, 0) = -9, ƒyy(0, 0) = -4, and ƒxy(0, 0) = -6
- If z = f (x, y) is a function that admits second continuous partial derivatives suchthat image 1 A critical point of f that generates a maximum point is: image 21. Identify where the local maximum occurs of the function (x,y). 2. Identify where the local minimum of the function occurs (x,y).Use partial derivatives to find the coordinates which give the shortest distance from the point (2,0,0) to the plane 3x+2y-z=2. Confirm that the point is the shortest distance with use of second derivatives
- Suppose ƒ is differentiable on (- ∞, ∞) andƒ(5.01) - ƒ(5) = 0.25. Use linear approximation to estimate the value of ƒ'(5).Find the maximum or local minimum f(x)=sinx.[0,4]Find the absolute minimum and absolute maximum of f(x,y)=19−9x+13y on the closed triangular region with vertices (0,0),(13,0) and (13,15). Minimum value: Occurs at: Maximum value: Occurs at: